Coherent skew mirrors

ABSTRACT

Systems and methods for performing coherent diffraction in an optical device are disclosed. An optical device may include a grating medium with a first hologram having a first grating frequency. A second hologram at least partially overlapping the first hologram may be provided in the grating medium. The second hologram may have a second grating frequency that is different from the first grating frequency. The first and second holograms may be pair-wise coherent with each other. A manufacturing system may be provided that writes the pair-wise coherent holograms in a grating medium using a signal beam and a reference beam. Periscopes may redirect portions of the signal and reference beams towards a partial reflector, which combines the beams and provides the combined beam to a detector. A controller may adjust an effective path length difference between the signal and reference beams based on a measured interference pattern.

This application is a continuation of U.S. patent application Ser. No.16/144,390, filed Sep. 27, 2018, which claims the benefit of U.S.provisional patent application No. 62/580,317, filed on Nov. 1, 2017,each of which are hereby incorporated by reference herein in theirentireties.

BACKGROUND

The present disclosure relates generally to optical devices, includingoptical devices having sets of pair-wise coherent grating structures.

Conventional dielectric mirrors are produced by coating a surface(typically glass) with layers of materials that differ from each otherin their electric permittivity. The layers of materials are typicallyarranged so that Fresnel reflections from layer boundaries reinforceconstructively, producing large net reflectivity. Broadband dielectricmirrors can be designed by ensuring that this condition obtains over arelatively broad specified range of wavelengths and incidence angles.However, because the layers are deposited on a surface, the reflectiveaxis of a dielectric mirror is necessarily coincident with surfacenormal, i.e., the reflective axis is perpendicular to the mirrorsurface. Because of this constraint on the reflective axis, a dielectricmirror is disposed in some devices in a configuration that is suboptimalfor purposes other than reflection. Similarly, the reflective axis beingconstrained to surface normal makes a dielectric mirror entirelyinadequate for some purposes. Moreover, glass dielectric mirrors tend tobe relatively heavy, making them suboptimal or inappropriate forapplications requiring a relatively lightweight reflective component.

Conversely, conventional grating structures can reflect light about areflective axis that differs from surface normal of the medium in whichthe grating structure resides. However, for a given angle of incidence,angles of reflection for conventional grating structures typicallyco-vary with wavelength of incident light. Thus, using a conventionalgrating structure to reflect light avoids the constraint inherent indielectric mirrors that reflective axis coincide with surface normal.However, where a substantially constant reflective axis is required, aconventional grating structure is substantially limited to a singlewavelength (or very narrow range of wavelengths) for a given angle ofincidence. Similarly, a conventional grating structure is limited to asingle angle of incidence (or very narrow range of incidence angles), inorder to reflect light of a specified wavelength about a constantreflective axis.

Accordingly, requirements for a relatively simple device that reflectslight about a reflective axis not constrained to surface normal, andwhose angle of reflection for a given angle of incidence is constant atmultiple wavelengths, are not met by currently available reflectivedevices comprising either reflective grating structures or dielectricmirrors. A need therefore exists for such a reflective device, and suchneed may be acute in head mounted display devices.

SUMMARY

The described features generally relate to one or more improved methods,systems, or devices for performing diffraction using coherent skewmirrors that include pair-wise coherent holograms. The holograms may beimplemented within optical media as holographic optical elements.

In some examples, an optical device is provided that includes a gratingmedium and a first hologram in the grating medium that has a firstgrating frequency. A second hologram at least partially overlapping thefirst hologram may be provided in the grating medium. The secondhologram may have a second grating frequency that is different from thefirst grating frequency. The first and second holograms may be pair-wisecoherent with each other. For example, when the first and secondholograms both receive a coherent probe beam, the first hologram maygenerate a first diffracted beam, and the second hologram may generate asecond diffracted beam that is in a fixed phase relationship with thefirst diffracted beam. A set of holograms such as these may be formed inthe grating medium, each having a respective grating frequency. Thefirst and second holograms may have adjacent grating frequencies amongthe set of holograms. If desired, a spectral peak of light diffracted bythe first hologram may overlap with a spectral null of light diffractedby the second hologram.

In order to manufacture pair-wise coherent holograms such as these, ahologram writing system may be provided that writes the pair-wisecoherent holograms in a grating medium using a signal beam and areference beam. A first light redirecting element such as a firstperiscope may redirect a portion of the signal beam towards a partialreflector. A second light redirecting element such as a second periscopemay redirect a portion of the reference beam towards the partialreflector. The partial reflector may generate a combined beam bycombining the portion of the signal beam and the portion of thereference beam. A detector may measure an interference patternassociated with the combined beam. A controller may adjust an effectivepath length difference between the signal beam and the reference beambased on the measured interference pattern. For example, the controllermay adjust a piezo-actuated phase shifting mirror that operates on agiven one of the signal and reference beams. The controller may includea servo controller that adjusts the effective path length to compensatefor physical vibrations detected by the detector. This process may beiterated to actively ensure the alignment and coherence of the signaland reference beams for writing the pair-wise coherent holograms in thegrating medium. Other manufacturing methods and systems may be used ifdesired.

BRIEF DESCRIPTION OF DRAWINGS

A further understanding of the nature and advantages of implementationsof the present disclosure may be realized by reference to the followingdrawings. In the appended figures, similar components or features mayhave the same reference label. Further, various components of the sametype may be distinguished by following the reference label by a dash anda second label that distinguishes among the similar components. If onlythe first reference label is used in the specification, the descriptionis applicable to any one of the similar components having the same firstreference label irrespective of the second reference label.

FIG. 1A is a cross-section view illustrating reflective properties of anillustrative skew mirror in accordance with some embodiments.

FIG. 1B is a cross-section view illustrating reflective properties of anillustrative skew mirror in accordance with some embodiments.

FIG. 2A is a cross-section view illustrating reflective properties of anillustrative skew mirror in accordance with some embodiments.

FIG. 2B is a cross-section view illustrating reflective properties of anillustrative skew mirror in accordance with some embodiments.

FIG. 3 is a cross-section view of an illustrative system for making anillustrative skew mirror, in accordance with some embodiments.

FIG. 4 is a cross-section view illustrating an illustrative method ofmaking an illustrative skew mirror, in accordance with some embodiments.

FIG. 5A is a cross-section view of an illustrative hologram recorded ina grating medium in accordance with some embodiments.

FIG. 5B is a cross-section view of a k-space representation of anillustrative single sinusoidal hologram in accordance with someembodiments.

FIG. 6A is a cross-section view of a k-space representation of anillustrative single sinusoidal hologram in accordance with someembodiments.

FIG. 6B cross-section view of a k-space representation of anillustrative single sinusoidal hologram in accordance with someembodiments.

FIG. 7 is a cross-section real view illustrating reflective propertiesof an illustrative skew mirror in real space, in accordance with someembodiments.

FIG. 8A is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 8B is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 9A is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 9B is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 10A is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 10B is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 10C is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 10D is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 11A is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 11B is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 12A is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 12B is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 12C is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 13A is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 13B is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 14A is a cross-section view of a k-space representation of anillustrative skew mirror in accordance with some embodiments.

FIG. 14B is a cross-section view illustrating reflective properties ofan illustrative skew mirror in accordance with some embodiments.

FIG. 15 is a plan view illustrating reflective properties of anillustrative skew mirror in accordance with some embodiments.

FIG. 16A is a cross-section view illustrating an illustrative system formaking an illustrative skew mirror, in accordance with some embodiments.

FIG. 16B is a cross-section view illustrating an illustrative system formaking an illustrative skew mirror, in accordance with some embodiments.

FIG. 17 is a diagram of an illustrative skew mirror having pair-wisecoherent holograms in accordance with some embodiments.

FIG. 18 is a diagram showing a first hologram in an illustrative skewmirror may have a spectral peak aligned with a spectral null in anadjacent second hologram in the skew mirror in accordance with someembodiments.

FIG. 19 is a diagram of an illustrative system that may be aligned formaking an illustrative skew mirror having pair-wise coherent hologramsin accordance with some embodiments.

FIG. 20 is a diagram of an illustrative system that may be activelyaligned for making an illustrative skew mirror having pair-wise coherentholograms in accordance with some embodiments.

FIGS. 21 and 22 are perspective views of illustrative periscopes thatmay be used in a system of the type shown in FIG. 20 in accordance withsome embodiments.

FIG. 23 is a flow chart of illustrative steps that may be used inactively aligning a system of the type shown in FIG. 20 for making anillustrative skew mirror having pair-wise coherent holograms inaccordance with some embodiments.

DETAILED DESCRIPTION

Embodiments of the present disclosure include a diffractive deviceincluding a grating medium within which resides a hologram or othergrating structures. The grating medium, by virtue of the gratingstructures residing therein, has physical properties that allow it todiffract light about an axis, referred to as a reflective axis, whereinangle of diffraction (henceforth referred to as angle of reflection) issubstantially constant, (e.g. it varies by less than) 1° for multiplewavelengths of light incident upon the grating medium at a given angleof incidence. In some embodiments, the above phenomenon is observed formultiple angles of incidence. The grating structures may exhibitpair-wise coherence.

Similarly, embodiments typically have a substantially constantreflective axis across a range of incidence angles for incident light ofa given wavelength, and this phenomenon may be observed with incidentlight at various wavelengths. In some embodiments, the reflective axisremains substantially constant for every combination of a set ofmultiple incidence angles and a set of multiple wavelengths

In some embodiments, the grating structures includes a hologramgenerated by interference between multiple light beams referred to asrecording beams. Typically, but not necessarily, the grating structuresincludes multiple holograms. The multiple holograms may be recordedusing recording beams incident upon the grating medium at angles thatvary among the multiple holograms, and/or using recording beams whosewavelengths vary among the multiple holograms. In some embodiments, thegrating structures include a hologram recorded using two recording beamswhose angles of incidence upon the grating medium vary while thehologram is being recorded, and/or whose wavelengths vary while thehologram is being recorded. Embodiments further include a device whereinthe reflective axis differs from surface normal of the grating medium byat least 1.0 degree; or at least by 2.0 degrees; or at least by 4.0degrees; or at least by 9.0 degrees. The grating structures (e.g., a setof holograms in a grating medium) may sometimes be referred tocollectively herein as a skew mirror, as described in greater detailbelow. Two or more holograms in the skew mirror may be pair-wisecoherent (e.g., the skew mirror may be a coherent skew mirror).

A first embodiment skew mirror 100 is illustrated in FIGS. 1A and 1B.The first embodiment skew mirror 100 (sometimes referred to herein as aset of gratings or volume holographic grating structures) includes agrating structure 105 (shown by diagonal hatch lines in FIGS. 1A and 1B)residing in a grating medium 110. For purposes of clarity, the diagonalhatch lines are omitted in a region within the grating medium 110proximate figure elements indicating light, axes, and angles. However,grating structure 105 typically occupies the region described above. Thegrating structure 105 of the first embodiment includes multipleholograms that at least partially spatially overlap with each other inthe grating medium 110.

The multiple holograms are recorded into the grating medium internalvolume and thus extend below the grating medium surface 112.Accordingly, they are sometimes referred to as volume holograms. Themultiple holograms of the first embodiment comprise forty eight (48)volume holograms, recorded with recording beams having a wavelength of405 nm. Each of the 48 volume holograms typically at least partiallyspatially overlaps all others of the 48 volume holograms in the gratingmedium 110. In some embodiments, each of the multiple holograms at leastpartially spatially overlaps at least one, but not all, of the other ofthe multiple holograms. Recording the 48 holograms of the firstembodiment skew mirror is described below in a first method of making askew mirror. In some embodiments, the grating structure includes between1 and 48 holograms; or between 4 and 25 holograms; or at least 5holograms; or at least 9 holograms; or at least 11 holograms; or atleast 24 holograms.

The first embodiment grating medium 110 may be a recording medium thatis approximately 200 urn thick, has an M/# of approximately 18, and arefractive index of approximately 1.50 for 405 nm light, as just oneexample. Optical recording mediums such as this are a type of gratingmedium in which grating structures can be recorded by optical means.Grating mediums are typically, but not necessarily, at least 70 um thickto approximately 1.2 mm thick. The grating medium may undergo relativelylittle shrinkage (usually about 0.1% to 0.2%) as a result of recordingvolume holograms. Variations of grating mediums include, but are notlimited to, photorefractive crystals, dichromated gelatin,photo-thermo-refractive glass, and film containing dispersed silverhalide particles.

Variations of the first embodiment skew mirror 100 may include anadditional layer such as a glass cover or glass substrate (not shown inFIGS. 1A and 1B). The additional layer may serve to protect the gratingmedium from contamination, moisture, oxygen, reactive chemical species,damage, and the like. The additional layer is typically refractive indexmatched to the grating medium 110. Because the refractive index for theadditional layer is usually very close to the refractive index of thegrating medium, refraction of light at the interface of the additionallayer and the grating medium can usually be ignored. For the firstembodiment, refractive indices for both the additional layer and thegrating medium are approximately 1.5 for light having a wavelength of405 nm. For clarity, the additional layer is not shown in FIGS. 1A and1B.

As best seen in FIG. 1A, the grating structure 105 of the firstembodiment has the physical property of being configured to reflect afirst incident light 124A, 124B, about a first reflective axis 138(shown in broken line). The first incident light consists essentially ofa collimated, monochromatic light beam. The first incident lightfurthermore includes a first wavelength of 532 nm and is incident uponthe grating medium 110 at a specific site 117. The first reflective axis138 differs from surface normal 122 of the grating medium by a firstreflective axis angle 135 of +13.759 degrees (internal, relative tosurface normal), where the first incident light has an first internalangle of incidence 125A, 125B relative to surface normal, from −4.660degrees (shown as first incident light 124A) to +1.933 degrees (shown asfirst incident light 124B), resulting in a range of 6.593 degrees. Thefirst internal angles of incidence for the first incident light includeone hundred (100) different internal angles spaced at angle intervals ofabout 0.067 degrees, from −4.660 degrees to +1.933 degrees. In somevariations of the first embodiment skew mirror, the first internalangles of incidence for the first incident light include ten (10)different internal angles spaced at angle intervals of about 0.67degrees, from −4.660 degrees to +1.933 degrees. Throughout thisspecification and appended claims, identified angles and angle valuesrefer to internal angles relative to surface normal, unless clearlyindicated otherwise. These example angles are merely illustrative andnon-limiting.

As shown FIG. 1A, first incident light 124A, having a first internalangle of incidence of 125A of −4.660 degrees relative to surface normal,is reflected by the grating structure 105 as first reflected light 127A,having a first internal angle of reflection 126A of +32.267 degreesrelative to surface normal. First incident light 124B, having a firstinternal angle of incidence 125B relative to surface normal of +1.933degrees, is reflected as first reflected light 127B having a firstinternal angle of reflection 126B of +25.668 degrees. First reflectedlight 127A, 127B has the first wavelength, e.g., in the first embodimentthe first reflected light has a wavelength of 532 nm.

Incident light and its reflection are bisected by the reflective axissuch that the internal angle of incidence of the incident light relativeto the reflective axis has the same magnitude as the internal angle ofreflection of the reflected light relative to the reflective axis. Thusit can be said that the incident light and its reflection exhibitbilateral symmetry about the reflective axis.

As best seen in FIG. 1B, the grating structure 105 of the firstembodiment is further configured to reflect second incident light 130A,130B about a second reflective axis 139. The second incident light isessentially a collimated, monochromatic light beam. The second incidentlight furthermore includes a second wavelength of 513 nm and is incidentupon the grating medium 110 at the specific site 117. The specific site117 includes an area of the grating medium surface 112 upon which boththe first and second incident light shine. The second reflective axis139 differs from surface normal 122 of the grating medium by a secondreflective axis angle 136 of +13.693 degrees (internal) relative tosurface normal, where the second incident light has a second internalangle of incidence, relative to surface normal, from −4.660 degrees to+1.933 degrees. The second internal angle of incidence includes onehundred (100) different internal angles spaced at angle intervals ofapproximately 0.067 degrees, from −4.660 degrees to +1.933 degrees. Insome variations of the first embodiment skew mirror, the second internalangles of incidence for the second incident light include ten (10)different internal angles spaced at angle intervals of about 0.67degrees, from −4.660 degrees to +1.933 degrees. These angles are merelyillustrative and non-limiting.

As shown in FIG. 1B, second incident light 130A, having a secondinternal angle of incidence 128A of −4.660 degrees relative to surfacenormal, is reflected by the grating structure 105 as second reflectedlight 133A, having a second internal angle of reflection 133A of +32.075degrees relative to surface normal. Second incident light 130B, having asecond internal angle of incidence 128B relative to surface normal of+1.933 degrees, is reflected as second reflected light 133B having asecond internal angle of reflection 129B of +25.273 degrees. Secondreflected light 133A, 133B has the second wavelength, e.g., in the firstembodiment the second reflected light has a wavelength of 513 nm.

The first wavelength (532 nm) differs from the second wavelength (513nm) by 19 nm, which can be represented by a value referred to as a wavefraction (WF), defined as

${{WF} = \frac{\left( {{\lambda 1} - {\lambda 2}} \right)}{\left( {{\lambda 1} + {\lambda 2}} \right)/2}},$

where λ1=a longer wavelength among multiple wavelengths, and λ2=ashorter wavelength among the multiple wavelengths. Thus where themultiple wavelengths consist of a first wavelength of 532 nm and asecond wavelength of 513 nm,

${WF} = {\frac{\left( {532 - 513} \right)}{\left( {532 + 5} \right)\text{/}2} = {0.036.}}$

Similarly, where the multiple wavelengths consist of a continuousspectrum from 390 nm or less to at least 700 nm, WF≥0.57. Embodimentsinclude, but are not limited to, variations in which WF≥0.005; WF≥0.010;WF≥0.030; WF≥0.10; WF≥0.250; WF≥1.0; or WF≥2.0. The wave fraction (WF)defined by a longer (λ1) and shorter (λ2) wavelengths in the rangetypically, but not necessarily, includes a continuous spectrum ofwavelengths between λ1 and λ2.

The second reflective axis angle 136 differs from the first reflectiveaxis angle 135 by 0.0661 degree. Accordingly, the second reflective axisis substantially coincident with the first reflective axis, meaning thatthe second reflective axis angle 136 differs from first reflective axisangle 135 by 1.0 degree or less. Such small difference betweenreflecting axis angles across a range of wavelengths (in this case,across a WF of 0.039) can be a necessity where a nondispersive mirror isrequired. For some applications, the difference between reflective axisangles should be 0.250 degree or less for WF=0.030. Similarly, for someother applications, the difference between reflective axis angles shouldbe equal 0.10 degree or less for WF=0.030.

Relative to the first reflective axis, internal angles of incidence ofthe first incident light vary from −11.867 degrees to −18.464 degrees.Relative to the second reflective axis, internal angles of incidence ofthe second incident light vary from −11.670 degrees to −18.368 degrees.Thus it can be said that each of the first incident light and secondincident light is offset from the first reflective axis by at least11.670 degrees. In embodiments, incident light may be offset from itsreflective axis by an internal angle of at least 1.0 degree; by at least2.0 degrees; by at least 5.0 degrees; or by at least 9.0 degrees. A skewmirror or other reflective device configured to reflect incident lightthat is offset from the incident light's reflective axis can beadvantageous in some applications. For example, in a head mounteddisplay it may be advantageous to reflect an image toward a user's eye,but not to retro-reflect the image back toward its source. Suchreflection toward a user's eye typically requires that incident light beoffset from its reflective axis by an internal angle of at least 5.0degrees, and more typically by at least 9.0 degrees. Similarly, a deviceutilizing total internal reflection typically requires that incidentlight be offset from its reflective axis.

First embodiment external angles relative to surface normal for incidentlight and its reflection are also illustrated in FIGS. 1A and 1B. Asseen in FIG. 1A, external angles relative to surface normal for firstincident light 124A, 124B ranges from first incident light externalangle 113A of −7.000 degrees to first incident light external angle 113Bof +2.900 degrees. As seen in FIG. 1B, external angles relative tosurface normal for second incident light 130A, 130B ranges from secondincident light external angle 115A of −7.000 to second incident lightexternal angle 115B of +2.900 degrees. First reflected light externalangles 114A, 114B and second reflected light external angles 116A, 116Bare also illustrated in FIGS. 1A and 1B, respectively. External anglesare measured with the skew mirror residing in air, with refractionoccurring at the skew mirror/air boundary.

The physical properties of the first embodiment allow it to reflectlight having other wavelengths, and to reflect light incident upon thegrating medium at other angles. For example, the first embodimentgrating structure's reflective properties allow it to reflect lighthaving a wavelength of 520.4 nm about a reflective axis having a meanreflective axis angle of +13.726 degrees that varies by 0.10 degree orless where angles of incidence of the 520.4 nm light range from −6.862degrees to +13.726 degrees and all angles in between, for a range of20.588 degrees. In another example of its reflective properties, thefirst embodiment is configured to reflect incident light about areflective axis (having a mean reflective axis angle of +13.726°) thatvaries by 0.20 degree or less for all wavelengths from 503 nm to 537 nm(a range of 34 nm, WF=0.065, including a continuous spectrum ofwavelengths between 503 nm and 537 nm), where the angle of incidence(internal, relative to surface normal) is −1.174 degrees.

For clarity, light in FIGS. 1A and 1B is illustrated as being reflectedat a point residing proximate a center of the grating structure 105.However, light is typically reflected throughout the grating structurerather than at a specific point. In some embodiments, the first incidentlight and the second incident light have wavelengths other than 532 and513, respectively. Similarly, embodiments include first and secondreflective axes that may be coincident with surface normal, or maydiffer from surface normal. The example of FIGS. 1A and 1B is merelyillustrative.

As a second example, a second embodiment skew mirror 200 is illustratedin FIGS. 2A and 2B. The second embodiment skew mirror 200 includes agrating structure 205 (shown by diagonal hatch lines in FIGS. 2A and 2B)residing in a grating medium 210. For purposes of clarity, the diagonalhatch lines are omitted in a region within the grating medium 210proximate figure elements indicating light, axes, and angles. However,the grating structure 205 typically occupies the region described above.The grating structure 205 of the second embodiment includes multipleholograms that at least partially overlap with each other in the gratingmedium 210. The multiple holograms of the second embodiment compriseforty nine (49) volume holograms, recorded with recording beams having awavelength of 405 nm. The 49 volume holograms overlap each other in thegrating medium 210, and are recorded in a manner similar to the firstembodiment skew mirror, except that recording beam internal angles ofincidence are adjusted to account for media shrinkage. Recording the 49holograms of the second embodiment skew mirror is described below in asecond method of making a skew mirror.

The second embodiment grating medium 210 may be a photosensitivepolymeric optical recording medium that is approximately 200 um thick,has an M/# of approximately 24, and a refractive index of approximately1.50 for light having a wavelength of 405 nm. This medium may shrinksabout 0.50% as a result of recording volume holograms.

Variations of the second embodiment skew mirror 200 may include anadditional layer such as a glass cover or glass substrate (not shown inFIGS. 2A and 2B). The additional layer is typically refractive indexmatched to the grating medium, and a thin film of index matching fluidmay reside between the grating medium 210 and the additional layer.

As best seen in FIG. 2A, the grating structure 205 of the secondembodiment has the physical property of being configured to reflect afirst incident light 224A, 224B, about a first reflective axis 238(shown in broken line). The first incident light of the secondembodiment consists essentially of a collimated, monochromatic lightbeam. The first incident light furthermore includes a first wavelengthof 532 nm and is incident upon the grating medium 210 at a specific site217. The first reflective axis 238 differs from surface normal 222 ofthe grating medium by a first reflective axis angle 235 of +14.618degrees (internal) relative to surface normal, where the first incidentlight has a first internal angle of incidence 225A, 225B, relative tosurface normal, residing between −9.281 degrees to −2.665 degrees,inclusive (a range of 6.616 degrees). The first internal angle ofincidence includes one hundred one (101) different internal anglesspaced at angle intervals of approximately 0.066 degrees, from −9.281degrees to −2.665 degrees. In some variations of the second embodimentskew mirror, the first internal angles of incidence for the firstincident light include ten (10) different internal angles spaced atangle intervals of about 0.66 degrees, from −9.281 degrees to −2.665degrees. These angles are merely illustrative and non-limiting.

As shown FIG. 2A, first incident light 224A, having a first internalangle of incidence 225A of −9.281 degrees relative to surface normal, isreflected by the grating structure 205 as first reflected light 227A,having a first internal angle of reflectance 226A of +38.610 degreesrelative to surface normal. First incident light 224B, having a firstinternal angle of incidence 225B relative to surface normal of −2.665degrees, is reflected as first reflected light 227B having a firstinternal angle of reflectance 226B of +31.836 degrees. First reflectedlight 224A, 224B has the first wavelength, i.e. in the second embodimentthe first reflected light has a wavelength of 532 nm.

As best seen in FIG. 2B, the grating structure 205 of the secondembodiment is further configured to reflect second incident light 230A,230B about a second reflective axis 239. The second incident light ofthe second embodiment consists essentially of a collimated,monochromatic, light beam. The second incident light furthermoreincludes a second wavelength of 513 nm, and the second wavelengththerefore differs from the first wavelength by 19 nm, or a wave fraction(WF) of 0.036. The second incident light is incident upon the gratingmedium 210 at the specific site 217. The specific site 217 of the secondembodiment includes an area of the grating medium surface 212 upon whichboth the first and second incident light shine. The second reflectiveaxis 239 differs from surface normal 222 of the grating medium by asecond reflective axis angle 236 of +14.617 degrees (internal) relativeto surface normal, where the second incident light has a second internalangle of incidence 228A, 228B relative to surface normal, spanning arange of −9.281 degrees to −2.665 degrees. The second internal angle ofincidence of the second incident light includes one hundred one (101)different internal angles spaced at angle intervals of approximately0.066 degrees, from −9.281 degrees to −2.665 degrees. In some variationsof the second embodiment skew mirror, the second internal angles ofincidence for the second incident light include ten (10) differentinternal angles spaced at angle intervals of about 0.66 degrees, from−9.281 degrees to −2.665 degrees.

As shown in FIG. 2B, second incident light 230A, having a secondinternal angle incidence 228A of −9.281 degrees relative to surfacenormal, is reflected by the grating structure 205 as second reflectedlight 233A, having a second internal angle of reflectance 229A of+38.598 degrees relative to surface normal. Second incident light 230B,having a second internal angle of incidence 228B relative to surfacenormal of −2.655 degrees, is reflected as second reflected light 233Bhaving a second internal angle of reflectance 229B of +31.836 degrees.Second reflected light 233A, 233B has the second wavelength, e.g., inthe second embodiment the second reflected light has a wavelength of 513nm.

For clarity, light in FIGS. 2A and 2B is illustrated as being reflectedat a point residing proximate a center of the grating structure 205.However, light is typically reflected throughout the grating structurerather than at a specific point. In the second embodiment, the secondreflective axis angle differs from the first reflective axis angle byapproximately 0.0005 degree across WF=0.036. This very low level ofchange can approach the level of precision of instrumentation used tomeasure reflection angles. Accordingly, the second reflective axis canbe said to not differ from the first reflective axis. For someapplications, the difference between reflective axis angles should be0.025 degree or less. For some other applications, the differencebetween reflective axis angles should be 0.010 degree or less acrossWF≥0.036. The second embodiment skew mirror meets these requirements. AStudent's t-test (two-tailed) indicates no difference between the firstreflective axis angle and the second reflective axis angle (N=101 pergroup; P=0.873). Moreover, a difference of 0.001 degree or lesschallenges the precision of instrumentation used to measure skew mirrorreflection angles. Accordingly, for purposes of the present invention,where a second reflective axis differs from a first reflective axis by0.001 degree or less, the second reflective axis can be said to notdiffer from the first reflective axis.

For the second embodiment skew mirror, angles of incidence of the firstincident light vary from −17.250 degrees to −23.946 degrees relative tothe first reflective axis. Angles of incidence of the second incidentlight relative to the second reflective axis vary from −17.250 degreesto −23.940 degrees. Thus it can be said that each of the first incidentlight and second incident light is offset from the first reflective axisby at least 17.20 degrees. This is merely illustrative and non-limiting.

Second embodiment external angles relative to surface normal forincident light and its reflection are also illustrated in FIGS. 2A and2B. As seen in FIG. 2A, external angles relative to surface normal forfirst incident light 224A, 224B ranges from first incident lightexternal angle 213A of −14.000 degrees to first incident light externalangle 213B of −4.000 degrees. As seen in FIG. 2A, external anglesrelative to surface normal for second incident light 230A, 230B rangesfrom second incident light external angle 215A of −14.000 to secondincident light external angle 215B of −4.000 degrees. First reflectedlight external angles 214A, 214B and second reflected light externalangles 216A, 216B are also illustrated in FIGS. 2A and 2B, respectively.

Incident light and its reflection can typically be reversed, such thatwhat was previously an angle of reflection becomes and angle ofincidence, and vice versa. However, recitation or description ofincidence angles herein refers only to those incidence angles beingoriented to one side of the incidence angles' reflective axes, or, inthe case of retro-reflected incident light, an incidence angle of zero(0) relative to the reflective axis. Accordingly, a range of incidenceangles does not include angles that are both positive and negative withrespect to the reflective axes. As illustrated and described here,incidence angles are negative (i.e. in a clockwise direction) withrespect to the incident lights' reflective axes. However, thisconvention is used for convenience and simplicity and is not meant toteach, suggest, or imply that a skew mirror can only reflect lightresiding to one side of a reflective axis. The example of FIGS. 2A and2B is merely illustrative.

As a third example, a third embodiment skew mirror includes a gratingstructure residing in a grating medium, where the grating structurecomprises twenty one (21) volume holograms that overlap each other inthe grating medium. The third embodiment grating medium may be aphotosensitive polymeric optical recording medium that is approximately70 urn thick, and that shrinks by about 1.0% as a result of recordingvolume holograms. Accordingly, shrinkage compensation is typicallyemployed when recording volume holograms in the third embodiment gratingmedium. Shrinkage compensation is described below in the method ofmaking the third embodiment skew mirror.

Variations of the third embodiment skew mirror may include an additionallayer such as a glass cover or glass substrate. The additional layer istypically refractive index matched to the grating medium, and a thinfilm of index matching fluid may reside between the third embodimentgrating medium and the additional layer.

The grating structure of the third embodiment has the physical propertyof being configured to reflect a first incident light about a firstreflective axis. The first incident light has a first wavelength of 532nm and is incident upon the grating medium at a specific site. The firstreflective axis differs from surface normal of the grating medium by afirst reflective axis angle of +9.419 degrees (internal) relative tosurface normal, where the first incident light has an internal angle,relative to surface normal, residing between −6.251 degrees and +0.334degrees, inclusive (a range of 6.585 degrees). The internal angle of thefirst incident light includes multiple angles spanning a range ofapproximately 6.59 degrees, the multiple angles including one hundred(100) different internal angles spaced at angle intervals ofapproximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.

Third embodiment first incident light having an internal angle of −6.251degrees relative to surface normal, is reflected by the gratingstructure as first reflected light having an internal angle of +25.027degrees relative to surface normal. First incident light having aninternal angle relative to surface normal of +0.334 degrees is reflectedas first reflected light having an internal angle of +18.487 degrees.First reflected light has the first wavelength, i.e. in the thirdembodiment the first reflected light has a wavelength of 532 nm.

The grating structure of the third embodiment is further configured toreflect second incident light about a second reflective axis. The secondincident light has a second wavelength of 513 nm, and the secondwavelength therefor differs from the first wavelength by 19 nm, or awave fraction (WF) of 0.036. The second incident light is incident uponthe grating medium at the specific site. The second reflective axisdiffers from surface normal of the grating medium by a second reflectiveaxis angle of +9.400 degrees (internal) relative to surface normal,where the second incident light has in internal angle, relative tosurface normal, sparming a range from −6.251 degrees to +0.334 degrees.The internal angle of the second incident light includes one hundred(100) different internal angles spaced at angle intervals ofapproximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.

Third embodiment second incident light, having an internal angle of−6.251 degrees relative to surface normal, is reflected by the gratingstructure as second reflected light, having an internal angle of +24.967degrees relative to surface normal. Second incident light having aninternal angle relative to surface normal of +0.334 degrees is reflectedas second reflected light having an internal angle of +18.425 degrees.Second reflected light has the second wavelength, i.e. in the thirdembodiment the second reflected light has a wavelength of 513 nm. Thesecond reflective axis of the third embodiment is substantiallycoincident with the first reflective axis.

Tables 1 includes a summary of reflective properties of first, second,and third embodiment skew mirrors.

TABLE 1 DIFFERENCE BETWEEN REFLECTIVE AXIS ANGLES AT λ = 532 nm AND λ =513 nm FIRST SECOND THIRD EMBODIMENT EMBODIMENT EMBODIMENT SKEW MIRRORSKEW MIRROR SKEW MIRROR (AK174-200 (AK233-200 (BAYFOL ® HX recordingmedium) recording medium) recording medium) N = 100 N = 101 N = 100measurements measurements measurements Mean reflective axis 13.693°14.617°  9.400° INTERNAL angle at λ = 532 nm * Mean reflective axis13.759° 14.618°  9.419° INTERNAL angle at λ = 513 nm * Differencebetween  0.066° 0.0005°  0.018° reflective axis INTERNAL angle at λ =532 nm and at λ = 513 nm** Incident Light −4.660° to +1.933° −9.281° to−2.665° −6.251° to +0.334° INTERNAL Angles *** (range = 6.593°) (range =6.616°) (range = 6.58520 ) Mean reflective axis 22.234° 25.594° 14.720°EXTERNAL angle at λ = 532 nm * Mean reflective axis 22.110° 25.593°14.690° EXTERNAL angle at λ = 513 nm * Difference between  0.124°0.0005°  0.030° reflective axis EXTERNAL angle at λ = 532 nm and at λ =513 nm** Incident Light −7.000° to 2.900° −14.000° to −4.000° −9.400° to+0.501° EXTERNAL Angles *** * mean angles are relative to surfacenormal, and are the means of N measurements at N incident light anglesof incidence; both incident and reflected light have the specifiedwavelength (λ). **differences between mean reflective axis angles at λ =532 nm and at λ = 513 nm are absolute values and thus excludes negativenumbers. *** incident light angles of incidence, relative to surfacenormal.

These examples are merely illustrative and, in general, the skew mirrorsdescribed herein may have other properties. In general, the skew mirror,sometimes be referred to herein as volume holographic grating structure,may have physical properties (by virtue of the grating structures orholograms therein) that allow it to diffract light about an axis,referred to as a reflective axis, wherein angle of diffraction(henceforth referred to as angle of reflection) varies by less than 1°for multiple wavelengths of light incident upon the grating medium at agiven angle of incidence. In some cases, the reflective axis is alsoconstant for multiple wavelengths and/or angles of incidence. In somecases, the grating structure is formed by one or more holograms. The oneor more holograms can be volume-phase holograms in some implementations.Other types of holograms may also be used in various implementations ofthe grating structure.

Each grating structure (e.g., each volume hologram) in the skew mirrormay reflect light in a manner different from another grating structurein the skew mirror. For example, a first grating structure may reflectincident light of a first wavelength at a first incidence angle, whereasa second grating structure may reflect incident light of a secondwavelength at the first incidence angle (e.g., different gratingstructures may be configured to reflect different wavelengths of lightfor incident light of the same incidence angle). Also, a first gratingstructure may reflect incident light of a first wavelength at a firstincidence angle, whereas a second grating structure may reflect incidentlight of the first wavelength at a second incidence angle (e.g.,different grating structures may be configured to reflect the samewavelength of light for incident light of different incidence angles).Furthermore, a grating structure may reflect first incident light of afirst wavelength and first incidence angle, and the grating structuremay reflect second incident light at a second wavelength and secondincidence angle about the same reflective axis. In this manner,different grating structures can be used to selectively reflect aparticular wavelength of light for incident light at a given incidenceangle. These different grating structures may be super-imposed withinthe grating medium of the skew mirror. The skew mirror may have asubstantially constant (uniform) reflective axis (e.g., each gratingstructure of the skew mirror has a same substantially constantreflective axis).

An exemplary system 350 for making a skew mirror is illustrated in FIG.3. The exemplary system 350 includes a grating medium 310 disposedbetween a first mirror 352A and a second mirror 352B. The first andsecond mirrors are arranged to direct a first recording beam 354 and asecond recording beam 355 such that the recording beams intersect andinterfere with each other to form an interference pattern that isrecorded as a hologram 305 in the grating medium 310. The hologram 305is an example of a grating structure.

The recording beams may sometimes be referred to as a reference beam anda signal beam. However, each of the first and second recording beams aretypically monochromatic collimated plane wave beams that are identicalto each other (except for angles at which they are incident upon thegrating medium). Moreover, the so-called signal beam typically includesno data encoded therein that is not also present in the so-calledreference beam. Thus designation of one recording beam as a signal beamand the other recording beam as a reference beam can be arbitrary, withthe designation of “signal” and “reference” serving to distinguishbetween the two recording beams, rather than to indicate that the onerecording beam includes encoded data not present in the other recordingbeam. In some embodiments the recording beams may have widths thatdiffer from each other.

The grating medium 310 is typically secured in place between a firstprism 359A and second prism 359B using a fluid index matched to both theprisms and the grating medium. A skew axis 361 resides at a skew angle364 relative to surface normal 322. The first and second recording beams354, 355 reside at a first recording beam internal angle 356 and asecond recording beam internal angle 357, respectively, relative surfacenormal 322. As can be seen in FIG. 3, the first and second recordingbeams 354, 355 are symmetrical about the skew axis 361 such that thefirst recording beam internal angle relative to the skew axis 366 isequal to 180° minus the second recording beam internal angle relative tothe skew axis 367. The internal angles of the first and second recordingbeams relative to the skew axis 366, 367 are readily calculated from thefirst and second recording beam internal angles 356, 357, respectively,and the skew angle 364.

Each of the first and second recording beams are typically collimatedplane wave beams originating from a laser light source. The plane wavebeams may be illustrated using multiple light ray depictions for eachrecording beam. For clarity however, in FIG. 3 the first and secondrecording beams are illustrated using a single light ray depiction foreach recording beam.

Refraction at air/prism boundaries, for example where the firstrecording beam 354 intersects an air/prism boundary of the first prism359A and where the second recording beam 355 intersects an air/prismboundary of the second prism 359B, is shown figuratively rather thanstrictly quantitatively in FIG. 3. Because the prisms are typicallyindex matched to the grating medium 310, refraction at the prism/gratingmedium boundary can usually be ignored. In embodiments, the gratingmedium and prisms each have an index of refraction of approximately1.50.

For purposes of the present invention, a skew angle can be substantiallyidentical to a reflective axis angle, meaning the skew angle is within1.0 degree of the reflective axis angle. The skew axis angle andreflective axis angle can theoretically be identical. However, due tolimits in system precision and accuracy, shrinkage of recording mediumthat occurs during recording holograms, and other sources of measurementerror, the skew angle as measured or estimated based on recording beamangles may not perfectly match the reflective axis angle as measured byincidence angles and reflection angles of light reflected by a skewmirror. Nevertheless, a skew angle determined based on recording beamangles can be within 1.0 degree of the reflective axis angle determinedbased on angles of incident light and its reflection, even where mediumshrinkage and system imperfections contribute to errors in estimatingskew angle and reflective axis angle. A skew axis/reflective axis isgenerally called a skew axis when referring to making a skew mirror (forexample when describing recording a hologram in a skew mirror gratingstructure), and as a reflective axis when referring to light reflectiveproperties of a skew mirror.

Angles at which the first and second recording beams 354, 355 areincident upon the grating medium are adjusted by rotating the first andsecond beam mirrors, 352A, 352B, respectively. Rotation of the beammirrors, indicated by rotation arrows 353, not only adjusts incidenceangles, but also would change where the recording beams intersect thegrating medium 310. Accordingly, the grating medium 310 and prisms 359A,359B are moved translationally in order to record holograms atapproximately the same location in the grating medium. Translation ofthe grating medium 310 is indicated by translation arrow 360.

In a variation of the exemplary system 350, a variable wavelength laseris used to vary the wavelength of the first and second recording beams.Incidence angles of the first and second recording beams may be, but arenot necessarily, held constant while the wavelength of the first andsecond recording beams is changed.

A first method of making a skew mirror is illustrated in FIG. 4. In theexample of FIG. 4, the skew mirror of the first method is the firstembodiment skew mirror 100, which is also illustrated in FIGS. 1A and1B, and whose physical properties are described above. The first methodtypically utilizes a system for making a skew mirror such as theexemplary system 350 illustrated in FIG. 3 and described above. Forclarity however, in FIG. 4 first and second prisms are omitted, andrecording beams are illustrated without showing refraction atair/grating medium boundaries or air/prism boundaries. However,refraction typically occurs at an air/prism boundary (or air/gratingmedium boundary, where index matched prisms are not used), and should beaccounted for when designing a system or method to achieve the internalangles described.

A first recording beam 154 and a second recording beam 155 are directedat the first embodiment grating medium 110, where the recording beamsinterfere with each other to create an interference pattern, which isrecorded as a volume hologram in the grating medium 110. The recordingbeams are typically created by splitting a 405 nm light beam (or lightof other wavelengths) from an external cavity, tunable diode laser intotwo separate beams. The light beam is split using a polarizing beamsplitter, and a half wave plate is used to alter polarity of one of thetwo separate beams from p-polarized to s-polarized, such that both ofthe two separate beams are s-polarized. One of the s-polarized beamsbecomes the first recording beam 154 and the other of the s-polarizedbeams becomes the second recording beam 155. Each of the first andsecond recording beams is a collimated, plane wave beam having awavelength of 405 nm.

The first embodiment skew mirror benefits from having reflectiveproperties that allow it to reflect light at a substantially differentwavelength, and in particular a considerably longer wavelength, than therecording beam wavelength. The grating medium of the first embodimentskew mirror, in which first embodiment holograms are recorded with 405nm wavelength recording beams, absorbs 405 nm light at approximately0.07 absorbance units for the 200 um thick medium, for example.Conversely, the grating medium has negligible absorbance for visiblewavelengths of light greater than 425 nm (conservatively estimated atless than 0.002 absorbance units per 200 um; the negligible absorbanceis typically indistinguishable from zero). Thus the grating mediumabsorbs recording beam light (at 405 nm) at least 35 times more stronglythan green light (for example, in a range of 503 nm to 537 nm) the firstembodiment skew mirror is configured to reflect.

The grating structure 105 of the first embodiment skew mirror 100 iscreated by recording 48 volume holograms in the grating medium 110. Eachof the 48 holograms is recorded at its own unique first recording beaminternal angle 156 and its own unique second recording beam internalangle 157. The first recording beam internal angle 156 is an internalangle of the first recording beam 154 relative to surface normal 122 ofthe grating medium 110 and the second recording beam internal angle 157is an internal angle of the second recording beam 155 relative tosurface normal 122. Each of the first and second recording beams for thefirst embodiment skew mirror has irradiance of approximately 3 mW/cm².Typically, the first of the 48 holograms is recorded with an energy doseof 35 mJ/cm², and the dose is increased by about 1.5% for eachsubsequent hologram. The total energy dose for recording all 48holograms is typically about 2.5 J/cm². Irradiance and energy dosesdescribed here are merely exemplary. Other embodiments of skew mirrorsand methods of making skew mirrors may use different levels ofirradiance and energy dose.

A first hologram is recorded using a first recording beam internal angle156 of +53.218 degrees and a second recording beam internal angle 157 of+154.234 degrees. The skew axis 164 has a skew angle 164 of +13.726degrees relative to surface normal 122. For each subsequent hologram ofthe grating structure, the first and second recording beam internalangles 156, 157 are typically changed by amounts that are approximatelyequal in magnitude to each other, but having opposite signs. Forexample, for a second hologram, the first recording beam internal angleis changed by +0.091 degree and the second recording beam internal angleis adjusted by −0.091 degree, such that the first recording beaminternal angle 156 becomes +53.309 degrees and the second recording beaminternal angle+154.143 degrees. The magnitudes of changes in recordingbeam internal angles from one hologram to the next hologram varyslightly across the 48 volume holograms (i.e. the change in change inrecording beam internal angles from one hologram to the next varies),from 0.091 degree for changes in recording beam internal angles from thefirst hologram to the second hologram, to 0.084 degree for changes inrecording beam internal angles from the 47^(th) hologram to the 48^(th)hologram. However, for each change of first and second recording beaminternal angles, the magnitude of change is the same and the sign isopposite for each of the first and second beam angles. The first andsecond recording beam internal angles 156, 157 for the last (48^(th))hologram of the first embodiment grating structure 105 are +57.332 and+150.120 degrees, respectively. In some embodiments, the magnitude ofchange of the first recording beam internal angle may differ veryslightly from the magnitude of change of the second recording beaminternal angle in order to compensate for system imprecision, for Snelleffects, for dispersion, or for shrinkage of the grating medium thatresults from recording the holograms.

The first recording beam internal angle 156 ranges from +53.218 to+57.332 degrees (a range of 4.114 degrees) and the second recording beaminternal angle 157 ranges from +154.234 to +150.120 degrees (a range of4.114 degrees). As can be seen in FIG. 4, the first and second recordingbeams 154, 155 are symmetrical about the skew axis 161 such that theinternal angle of the first recording beam relative to the skew axis 166(+38.492 degrees for the first hologram) is equal to 180° minus theinternal angle of the second recording beam relative to the skew axis167 (+141.508 degrees for the first hologram) (180°-+141.508°=38.492degrees). The internal angles of the first and second recording beamsrelative to the skew axis 166, 167 are readily calculated from the firstand second recording beam internal angles 156, 157, respectively, andthe skew angle 164. After recording the 48 volume holograms, therecording medium is light cured.

In a variation of the first method of making a skew mirror, a hologramis created by continuously and synchronously adjusting the first andsecond recording beam internal angles while maintaining the symmetry ofthe first and second recording beams about the skew axis as describedabove. Accordingly, a single hologram is recorded while the firstrecording beam is scanned from a first recording beam internal angle of+53.218 degrees to a first recording beam angle of +57.332 degrees.Simultaneously, the second recording beam is scanned from a secondrecording beam internal angle of +154.234 degrees to +150.120 degrees.The single hologram is thus equivalent to the 48 discrete hologramsrecorded with 48 sets of unique first recording beam and secondrecording beam internal angles, and the total energy dose for recordingthe single hologram is typically about the same (2.5 J/cm²) as for the48 holograms.

These examples are merely illustrative and, in general, similarrecording methods may be used to manufacture any desired skew mirrors.For example, a second method of making a skew mirror is described belowas an illustrative example. The skew mirror of the second method is thesecond embodiment skew mirror 200, which is also illustrated in FIGS. 2Aand 2B, and whose physical properties are described above.

The second method is identical to the first method except that first andsecond recording beam internal angles are different than with the firstmethod, and the grating medium also differs from the first method. Likethe first embodiment, the second embodiment skew mirror benefits fromhaving reflective properties that allow it to reflect light at asubstantially different wavelength, and in particular a considerablylonger wavelength, than the recording beam wavelength.

The grating structure 205 of the second embodiment skew mirror 200 iscreated by recording 49 volume holograms in the grating medium 210. Eachof the 49 holograms of the second method is recorded at its own uniquefirst recording beam internal angle and its own unique second recordingbeam internal angle. The first recording beam internal angle is aninternal angle of the first recording beam relative to surface normal ofthe grating medium and the second recording beam internal angle is aninternal angle of the second recording beam relative to surface normal.Each of the first and second recording beams for the first embodimentskew mirror has irradiance of approximately 3 mW/cm². Typically, thefirst of the 49 holograms are recorded with an energy dose of 35 mJ/cm²,and the dose is increased by about 1.5% for each subsequent hologram.The total dose for recording all 49 holograms is typically about 2.5J/cm².

According to the second method, a first hologram is recorded using afirst recording beam internal angle of +55.913 degrees and a secondrecording beam internal angle of +153.323 degrees. The skew axis has askew angle of +14.618 degrees relative to surface normal. For eachsubsequent hologram of the grating structure, the first and secondrecording beam internal angles are typically changed by amounts that areapproximately equal in magnitude to each other, but having oppositesigns. For example, for recording a second hologram according to thesecond method, the first recording beam internal angle is changed by+0.095 degree and the second recording beam internal angle is adjustedby −0.095 degree, such that the first recording beam internal anglebecomes +56.008 degrees and the second recording beam internal angle+153.228 degrees. The magnitudes of changes in recording beam internalangles from one hologram to the next hologram typically vary slightlyacross the 49 volume holograms (i.e. the change in change in recordingbeam internal angles from one hologram to the next varies), from amagnitude of 0.095 degree for changes in recording beam internal anglesfrom the first hologram to the second hologram, to a magnitude of 0.087degree for changes in recording beam internal angles from the 48^(th)hologram to the 49^(th) hologram. However, the magnitude of change isthe same for each of the first and second recording beam internalangles, and the sign of the change is opposite for each of the first andsecond recording beam internal angles. The first and second recordingbeam internal angles for the last (49^(th)) hologram of the secondembodiment grating structure are +60.252 and +148.984 degrees,respectively. In some embodiments, the magnitude of change of the firstrecording beam internal angle may differ very slightly from themagnitude of change of the second recording beam internal angle in orderto compensate for factors such as system imprecision, Snell effects,dispersion, or shrinkage of the grating medium that results fromrecording the holograms.

Thus according to the second method the first recording beam internalangle ranges from +55.913 to +60.252 degrees (a range of 4.339 degrees)and the second recording beam internal angle ranges from +153.323 to+148.984 degrees (a range of 4.339 degrees). As with the first method,the first and second recording beams of the second method aresymmetrical about the skew axis such that the internal angle of thefirst recording beam relative to the skew axis (+41.295 degrees for thefirst hologram) is equal to 180° minus the internal angle of the secondrecording beam relative to the skew axis (+138.705 for the firsthologram) (180°-+138.705°=+41.295 degrees). According to the secondmethod, the internal angles of the first and second recording beamsrelative to the skew axis are readily calculated from the first andsecond recording beam internal angles respectively, and the skew angle.After recording the 49 volume holograms, the recording medium is lightcured.

In a variation of the second method of making a skew mirror, a hologramis created by continuously and synchronously adjusting the first andsecond recording beam internal angles while maintaining the symmetry ofthe first and second recording beams about the skew axis as describedabove. Accordingly, a single hologram is recorded while the firstrecording beam is scanned from a first recording beam internal angle of+55.913 degrees to a first recording beam angle of +60.252 degrees.Simultaneously, the second recording beam is scanned from a secondrecording beam internal angle of +153.323 degrees to +148.984 degrees.The single hologram is thus equivalent to the 49 discrete hologramsrecorded with 49 sets of unique first recording beam and secondrecording beam internal angles. The total energy dose for recording thesingle hologram is typically 2.5 J/cm² for the single hologram. Theseexamples are merely illustrative and, in general, similar recordingmethods may be used to manufacture any desired skew mirrors.

As another example, in a multi-wavelength method of making a skewmirror, six volume holograms may be recorded in a grating medium, witheach of the six holograms being recorded using its own unique first andsecond recording beam internal angles of incidence. In addition, foreach of the six volume holograms, wavelengths of the first and secondrecording beams are adjusted continuously and synchronously from 403 nmto 408 nm, using a variable wavelength laser. Wavelengths of the firstand second recording beams are kept equal to each other while recordingeach of the six volume holograms. Total energy dose delivered inrecording the six volume holograms according to the multi-wavelengthmethod is typically, but not necessarily, 2.5 J/cm² for first and secondrecording beam internal angles of incidence for the multi-wavelengthmethod of making a skew mirror are provided below in Table 2. A skewmirror made by the multi-wavelength method has the same reflectivecharacteristics of the second embodiment skew mirror described above.

TABLE 2 RECORDING BEAM ANGLES FOR A MULTIWAVELENGTH METHOD OF MAKING ASKEW MIRROR First Recording Beam Second Recording Beam HOLOGRAM Angle ofIncidence* Angle of Incidence* 1 56.235° 153.001° 2 57.033° 152.203° 357.813° 151.423° 4 58.568° 150.668° 5 59.303° 149.933° 6 60.018°149.218° *internal, relative to grating medium surface normal

These examples are merely illustrative and, in general, similarrecording methods may be used to manufacture any desired skew mirrors(e.g., having any desired number of holograms for diffracting inputlight at particular incident angles and wavelengths in particulardirections, e.g., with a substantially constant reflective axis).Embodiments of a skew mirror can be created in a grating mediumincluding a volumetric dielectric medium, such as a photosensitiverecording medium. Skew mirror embodiments may be formed by constraininga spatial dielectric modulation spectrum as described herein. In anembodiment, dielectric modulation is accomplished holographically byrecording an interference pattern of two or more coherent light beams ina photosensitive recording medium. In other embodiments, dielectricmodulation can be accomplished by other means.

The k-space formalism is a method for analyzing holographic recordingand diffraction [1]. In k-space, propagating optical waves and hologramsare represented by three dimensional Fourier transforms of theirdistributions in real space. For example, an infinite collimatedmonochromatic reference beam can be represented in real space andk-space by equation (1),

E r ⁡ ( r ⇀ ) = A r ⁢ ⁢ exp ⁡ ( i ⁢ ⁢ k ⇀ r · r ⇀ ) ⁢ ⁢ → ⁢ ⁢ E r ⁡ ( k ⇀ ) = A r ⁢ ⁢δ ⁡ ( k ⇀ - k ⇀ r ) , ( 1 )

where E_(r)(

) is the optical scalar field distribution at all

={x,y,z} 3D spatial vector locations, and its transform E_(r)(

) is the optical scalar field distribution at all

={k_(r),k_(y),k_(z)} 3D spatial frequency vectors. A_(r) is the scalarcomplex amplitude of the field; and

is the wave vector, whose length indicates the spatial frequency of thelight waves, and whose direction indicates the direction of propagation.In some embodiments, all beams are composed of light of the samewavelength, so all optical wave vectors must have the same length, i.e.,

=k_(n). Thus, all optical propagation vectors must lie on a sphere ofradius k_(n)=2π n₀/λ, where n₀ is the average refractive index of thehologram (“bulk index”), and λ is the vacuum wavelength of the light.This construct is known as the k-sphere. In other embodiments, light ofmultiple wavelengths may be decomposed into a superposition of wavevectors of differing lengths, lying on different k-spheres.

Another important k-space distribution is that of the hologramsthemselves. Volume holograms usually consist of spatial variations ofthe index of refraction within a grating medium. The index of refractionspatial variations, typically denoted Δn(

), can be referred to as index modulation patterns, the k-spacedistributions of which are typically denoted Δn(

). The index modulation pattern created by interference between a firstrecording beam and a second recording beam is typically proportional tothe spatial intensity of the recording interference pattern, as shown inequation (2),

Δn(

)∝|E ₁(

)+E ₂(

)|² =|E ₁(

)|² +|E ₂(

)|² +E* ₁(

)E ₂(

)+E ₁(

)E* ₂(

)  (2)

where E₁ (

) is the spatial distribution of the signal first recording beam fieldand E₂(

) is the spatial distribution of the second recording beam field. Theunary operator * denotes complex conjugation. The final term in equation(2), E₁(

)E*₂(

), maps the incident second recording beam into the diffracted firstrecording beam. Thus we can write equation (3),

E 1 ⁡ ( r ⇀ ) ⁢ E 2 * ⁡ ( r ⇀ ) ⁢ ⁢ → ⁢ ⁢ E 1 ⁡ ( k ⇀ ) ⊗ E 2 ⁡ ( k ⇀ ) , ( 3 )

where ⊗ is the 3D cross correlation operator. This is to say, theproduct of one optical field and the complex conjugate of another in thespatial domain becomes a cross correlation of their respective Fouriertransforms in the frequency domain.

FIG. 5A illustrates a real space representation of recording a hologram505 in a grating medium 510 using a second recording beam 515 and afirst recording beam 514. The grating medium typically includes arecording layer configured to record interference patterns as holograms.FIG. 5A omits grating medium components other than the recording layer,such as an additional layer that might serve as a substrate orprotective layer for the recording layer. The second recording beam 515and first recording beam 514 are counter-propagating. Each of the secondrecording beam 515 and first recording beam 514 are typically plane wavebeams having the same wavelength as each other, and the first recordingbeam 514 typically contains no information encoded therein that is notalso present in the second recording beam. Thus the first and secondrecording beams, which can be referred to as signal and reference beams,are typically substantially identical to each other except for angles atwhich they are incident upon the recording medium 510.

FIG. 5B illustrates a k-space representation of the first and secondrecording beams, and the hologram. The hologram illustrated in FIGS. 5Aand 5B is a simple Bragg reflection hologram generated with thecounter-propagating first recording beam 514 and second recording beam515, and recorded in recording medium 510. FIG. 5A shows the secondrecording beam 515 and the first recording beam 514 impinging onopposite sides of the grating medium 510. Optical scalar fielddistributions at all

={x, y, z} 3D spatial vector locations for each of the second recordingbeam 515 and the first recording beam 514 can be represented as E₂(

) and E₁(

), respectively. The recording beams 514, 515 form planar interferencefringes, which are recorded as a hologram 505 within the grating medium510. The hologram 505 comprises a sinusoidal refractive index modulationpattern, and can be represented as Δn(

). In a counter-propagating configuration, the recorded planarinterference fringes have a spacing exactly half that of the (internal)wavelength of the light used to record the hologram.

FIG. 5B shows a k-space representation of the situation illustrated inreal space by FIG. 5A. The recording beams are represented in FIG. 5B bypoint-like k-space distributions lying on opposite sides of therecording k-sphere 570. As illustrated in FIG. 5B, the second recordingbeam has a k-space distribution 562, and the first recording beam has ak-space distribution 563. The second recording beam k-space distribution562 can be represented as E₂(

) and the first recording beam k-space distribution 563 can berepresented as E₁(

). Each of the second recording beam k-space distribution 562 and thefirst recording beam k-space distribution 563 are “point-like.” Secondrecording beam wave vector 564 and first recording beam wave vector 565,are shown extending from the origin to the second recording beam k-spacedistribution 562 and first recording beam k-space distribution 563,respectively. The second recording beam wave vector 564 can berepresented as E₂(

) and the first recording beam wave vector 565 can be represented as E₁(

). The hologram itself is represented in FIG. 5B by two conjugatesideband k-space distributions 568, each of which can be represented asΔn(

) and referred to as a Δn(

) k-space distribution. The two Δn(

) k-space distributions 568 have a small, finite size, but are“point-like” in the sense that they are typically several orders ofmagnitude smaller than their distance to the origin, or other featuresof FIG. 5B. For instance, if the thickness of grating medium 510 is 200μm with refractive index 1.5 and the recording beams have a wavelengthof 532 nm, then distributions 568 each resemble a sinc function alongthe k_(z) dimension with size 3.14×10⁴ rad/m null-to-null. However,their distance from the origin is 3.56×10⁷ rad/m, which is more than1000 times as large. Unless specified otherwise, all recited wavelengthsrefer to vacuum wavelengths.

Typically, the hologram constitutes a refractive index distribution thatis real-valued in real space. Locations of the two Δn(

) k-space distributions 568 of the hologram may be determinedmathematically from the cross-correlation operations E₂(

)⊗E₁(

) and E₁(

)⊗E₂(

), respectively, or geometrically from vector differences

_(G+)=

₁−

₂ and

_(G−)=

₂−

₁, where

_(G+) and

_(G−) are grating vectors from the respective hologram Δn(

) k-space distributions to the origin (not shown individually). Acombined grating vector 569, which can be represented as

_(G), comprising both

_(G+) and

_(G−) grating vectors, is shown in FIG. 5B as double headed arrow 569extending between the second recording beam k-space distribution 562 andthe first recording beam k-space distribution 563. Note that byconvention, wave vectors are represented by a lowercase “k,” and gratingvectors by uppercase “K.”

Once recorded, the hologram may be illuminated by a probe beam toproduce a diffracted beam. For purposes of the present invention, thediffracted beam can be considered a reflection of the probe beam, whichcan be referred to as an incident light beam. The probe beam and itsreflected beam are angularly bisected by a reflective axis (e.g. theangle of incidence of the probe beam relative to the reflective axis hasthe same magnitude as the angle of reflection of the reflected beamrelative to the reflective axis). The diffraction process can berepresented by a set of mathematical and geometric operations in k-spacesimilar to those of the recording process. In the weak diffractionlimit, the diffracted light distribution of the diffracted beam is givenby equation (4),

E _(d)(

)∝Δn(

)*E _(p)(

)

  (4)

where E_(d) (

) and E_(p)(

) are k-space distributions of the diffracted beam and the probe beam,respectively; and “*” is the 3D convolution operator [1]. The notation “

” indicates that the preceding expression is evaluated only where |

|=k_(n), i.e., where the result lies on the k-sphere. The convolutionΔ_(n)(

)*E_(p)(

) represents a polarization density distribution, and is proportional tothe macroscopic sum of the inhomogeneous electric dipole moments of thegrating medium induced by the probe beam, E_(p)(

).

Typically, when the probe beam resembles one of the recording beams usedfor recording, the effect of the convolution is to reverse the crosscorrelation during recording, and the diffracted beam will substantiallyresemble the other recording beam used to record the hologram. When theprobe beam has a different k-space distribution than the recording beamsused for recording, the hologram may produce a diffracted beam that issubstantially different than the beams used to record the hologram. Notealso that while the recording beams are typically mutually coherent, theprobe beam (and diffracted beam) is not so constrained. Amulti-wavelength probe beam may be analyzed as a superposition ofsingle-wavelength beams, each obeying Equation (4) with a differentk-sphere radius.

FIGS. 6A and 6B illustrate cases of Bragg-matched and Bragg-mismatchedreconstructions, respectively, generated by illuminating the hologramdepicted in FIGS. 5A and 5B. In both the Bragg-matched andBragg-mismatched cases, the hologram is illuminated with a probe beamhaving a shorter wavelength than the recording beams used to record thehologram. The shorter wavelength corresponds to a longer wave vector.Accordingly, a probe k-sphere 572 has a greater radius than that of therecording k-sphere 570. Both the probe k-sphere 572 and the recordingk-sphere 570 are indicated in FIGS. 6A and 6B.

FIG. 6A shows a case where the probe beam is designed to produce adiffracted beam k-space distribution 575 (represented as E_(d) (

)) that is point-like and lies on the probe beam k-sphere 572. Thediffracted beam k-space distribution 575 is produced according to theconvolution of Equation (4). The probe beam has a k-space distribution576 (represented as E_(p)(

)) that is also point-like. In this case, the probe beam is said to be“Bragg-matched” to the hologram, and the hologram may producesignificant diffraction, even though the probe beam wavelength differsfrom the wavelength of the recording beams used to record the hologram.As best seen in FIG. 6A, the convolution operation may also berepresented geometrically by the vector sum

_(d)=

_(p)+

_(G+), where

_(d) represents a diffracted beam wave vector 577,

_(p) represents a probe beam wave vector 578, and

_(G+) represents a sideband grating vector 579.

FIG. 6A shows a k-space representation of a mirror-like diffraction(which can be referred to as a reflection) of the probe beam by thehologram, where the probe beam angle of incidence with respect to thek_(z) axis is equal to the diffracted beam angle of reflection withrespect to the k_(z) axis. FIG. 6B shows a k-space representation of aBragg-mismatched case, wherein a k-space polarization densitydistribution 580, which can be represented as Δn(

)*E_(p)(

), does not lie on the probe k-sphere 572, and thus no significantdiffraction of the probe beam occurs. This non-diffracted k-spacedistribution 580 in the Bragg-mismatched case illustrated in FIG. 6B issomewhat analogous to the diffracted beam k-space distribution 575 inthe Bragg-matched case illustrated in FIG. 6A, but k-space distribution580 should not be referred to as a diffracted beam k-space distributionbecause no significant diffraction of the probe beam occurs.

Comparing the Bragg-matched and Bragg-mismatched cases, it is evidentthat the hologram will only produce mirror-like diffraction over a verysmall range of input angles for a given probe wavelength, if at all.This range may be somewhat extended by over-modulating the hologram, orby using a very thin recording layer; but these steps may still not leadto mirror-like behavior over a larger range of wavelengths and angles.

FIGS. 5A, 5B, 6A, and 6B represent a reflection hologram constituted bya single sinusoidal grating. As illustrated, this hologram exhibitsmirror-like reflectivity in a narrow band of wavelengths and incidenceangles. Conversely, embodiments of the present disclosure exhibit novelmirror-like reflectivity across relatively broad ranges of wavelengthsand angles by creating a more complex grating structure comprisingmultiple gratings.

FIG. 7 shows a geometry illustrating the Bragg selectivity of a singlesinusoidal grating. Grating medium 710 contains a single sinusoidalgrating of thickness d which reflects incident light 724 of a singlewavelength, Xo, as principal reflected light 727. At the Bragg-matchedcondition, incident light 724 impinges at angle θ_(i), and reflects asreflected light 727 at angle θ_(r), both angles measured with respect tothe z axis. Incident light 724 and reflected light 727 also define areflective axis 738, about which the angular magnitudes of incidenceθ_(i)′ and reflection θ_(r)′ are equal. Reflective axis 738 is thus anangular bisector of incident light 724 and reflected light 727.

The sinusoidal grating of FIG. 7 will exhibit both angular andwavelength Bragg selectivity. If incident light 724 impinges atnon-Bragg-matched angle θ_(i)-Δθ_(i), the diffraction efficiency may bediminished compared to the Bragg-matched diffraction efficiency. Theselectivity of a sinusoidal grating may be characterized by its angularBragg selectivity, Δθ_(B), given by equation (5):

$\begin{matrix}{{\Delta\theta}_{B} = {\frac{\lambda\mspace{14mu}\cos\mspace{14mu}\theta_{r}}{n_{0}\mspace{14mu} d\mspace{14mu}{\sin\left( {\theta_{i} - \theta_{r}} \right)}}.}} & (5)\end{matrix}$

In a weakly-diffracting sinusoidal grating, the angle θ_(i)+Δθ_(B)represents the first null in the angular diffraction efficiency plot.The quantity Δθ_(B) can thus be said to represent the angular width ofthe sinusoidal grating in that diffraction can be greatly diminishedwhen the angle of incidence deviates from the Bragg-matched angle θ_(i)by more than several times Δθ_(B). Similarly, for a weakly-diffractingsinusoidal grating, the skilled artisan would expect a reflective axisto vary considerably for monochromatic incident light whose angle ofincidence varies by more than several times Δθ_(B).

Conversely, illustrative skew mirrors exhibit relatively stablediffraction and a substantially constant reflective axis angle forincident light whose angle of incidence varies by many times Δθ_(B).Some skew mirror embodiments exhibit a substantially constant reflectiveaxis angle across a range of incident light angles of incidence of20×Δθ_(B). In embodiments, reflective axis angles across a range ofincident light angles of incidence of 20×Δθ_(B) change by less than0.250 degree; or by less than 0.10 degree; or by less than 0.025 degree.As shown in Table 3 below, reflective axis angles across a range ofincident light angles of incidence of 20×Δθ_(B) for the first and secondembodiment skew mirrors described above change by less than 0.020 degreeeach of the first and second embodiment skew mirrors, at multiplewavelengths that differ from each other by WF≥0.036.

TABLE 3 CHANGE IN REFLECTIVE AXIS ANGLES ACROSS AN INCIDENCE ANGLE RANGEOF APPROXIMATELY 20 x Δθ_(B) Difference In Skew Mirror reflectiveIncident Light Embodiment λ * Axis Angles ** Angle Range *** Δθ_(B) ^(†)FIRST 532 nm 0.012° −3.167° to +0.369° 0.177° EMBODIMENT 513 nm 0.012°−3.111° to +0.313° 0.171° SKEW MIRROR (AK174-200 recording medium)SECOND 532 nm 0.019° −7.246° to −4.726° 0.126° EMBODIMENT 513 nm 0.016°−7.202° to −4.770° 0.122° SKEW MIRROR (AK233-200 recording medium) *wavelength of both incident and reflected light. ** difference inreflective axis angles (internal, relative to surface normal) forincident light having a change in angle of incidence of approximately 20x Δθ_(B). *** range of incident light angles of incidence (internal,relative to surface normal) approximately equal to 20 x Δθ_(B), forwhich the Difference In Reflective Axis Angles is reported in thistable. ^(†)Δθ_(B) is calculated for an incident light angle of incidenceat the midpoint of the Incident Light Angle Range reported in thistable.

Similarly, a sinusoidal grating may be characterized by its wavelengthBragg selectivity, Δλ_(B), given by equation (6):

$\begin{matrix}{{\Delta\lambda}_{B} = {\frac{\lambda_{0}^{2}\mspace{14mu}\cos\mspace{14mu}\theta_{r}}{2n_{0}^{2}\mspace{14mu} d\mspace{14mu}{\sin^{2}\left( {\theta_{i} - \theta_{r}} \right)}}.}} & (6)\end{matrix}$

Those skilled in the art will recognize that in a weakly-diffractingsinusoidal grating, the wavelength λ₀+Δλ_(B) represents the first nullin the wavelength diffraction efficiency plot. The quantity Δλ_(B) canthus be said to represent the wavelength width of the sinusoidal gratingin that no significant diffraction will occur when the incidentwavelength deviates from the Bragg-matched wavelength λ₀ by more thanseveral times Δλ_(B). Equations (5) and (6) apply to changes in angleand wavelength only, respectively, and changing both angle andwavelength simultaneously may result in another Bragg-matched condition.

A grating may also be characterized by its diffracted angle response.For a sinusoidal grating, the diffracted angle response may be expressedby

Δθ_(r) cos θ_(r)=−Δθ_(i) cos θ_(i).  (7)

The diffracted angle response expresses the change in the angle ofreflection, Δθ_(r), in response to small changes in the angle ofincidence, Δθ_(i). In contrast, a true mirror has an angle responseexpressed by equation (8):

Δθ_(r)=−Δθ_(i).  (8)

A device that has a diffracted angle response substantiallycharacterized by Equation (7) may be said to exhibit grating-likereflective behavior, whereas a device that has a diffracted angleresponse substantially characterized by Equation (8) may be said toexhibit mirror-like reflective behavior. A device exhibitinggrating-like reflective behavior will necessarily also exhibit areflective axis that changes with angle of incidence, unless thatreflective axis is normal to the device surface, in which case cosθ_(r)=cos θ_(i). Accordingly, requirements for a relatively simpledevice that reflects light about a reflective axis not constrained tosurface normal, and whose angle of reflection for angles of incidencespanning multiples of its angular Bragg selectivity is constant atwavelengths spanning multiples of its wavelength Bragg selectivity, maynot be met by a single sinusoidal grating.

FIG. 7 illustrates a device geometry in a reflective configuration.However, the preceding analysis also applies to device geometries intransmissive configurations and to device geometries in which one orboth beams are waveguided by total internal reflection within thedevice.

FIGS. 8A and 8B illustrate operation of a skew mirror in k-spaceaccording to an embodiment. FIG. 8A shows two Δn(

) k-space distributions 888 for a hologram recorded in a grating mediumand configured to produce multi-wavelength mirror-like diffractionaccording to an embodiment. As explained above with respect to a k-spacerepresentation of a prior art hologram shown in FIG. 5B, a Δn(

) k-space distribution can be represented as Δn(

). A red k-sphere 890, green k-sphere 892, and blue k-sphere 893 inFIGS. 8A and 8B indicate k-spheres corresponding to wavelengths of lightresiding in the red, green, and blue regions of the visible spectrum,respectively.

Instead of two Δn(

) k-space distributions constituting a single sinusoidal grating (andwhich therefore can be characterized as “point-like”), the Δn(

) k-space distributions 888 shown in FIG. 8A are situated along asubstantially straight line in k-space, and thus can be characterized as“line segment-like”. In some embodiments, line segment-like Δn(

) k-space distributions comprise continuously-modulated sub-segments ofa substantially straight line in k-space. In some embodiments, linesegment-like Δn(

) k-space distributions substantially consist of point-likedistributions situated along a substantially straight line in k-space.The line segment-like Δn(

) k-space distributions 888 are situated symmetrically about the origin,and thus may be realized as conjugate sidebands of a real-valuedrefractive index distribution in real space (represented as Δn(

)). In some embodiments, the modulation may include absorptive and/oremissive components, and thus may not exhibit conjugate symmetry ink-space. The complex amplitude of the distribution may be uniform, or itmay vary in amplitude and/or phase while still exhibiting substantiallymulti-wavelength mirror-like diffraction according to embodiments of thepresent invention. In an embodiment, the line segment-like Δn(

) k-space distributions are situated substantially along the k_(z) axis,which, by convention, is the thickness direction of a recording layer.

FIG. 8B illustrates a multi-wavelength mirror-like reflective propertyof the hologram. Illumination of the hologram by a collimated probe beamwith point-like k-space distribution 876 (represented as E_(p)(

)) results in a k-space polarization density distribution 880(represented as Δn(

)*E_(p)(

) according to Equation (4). Because the probe beam k-space distribution876 is point-like, polarization density distribution 880 resembles asimple translation of Δn(

) k-space distribution 888 from the origin to the tip of probe beam wavevector 878 (

_(p)). Then, also according to Equation (4), only the part of thek-space polarization density distribution 880 (Δn(

)*E_(p)(

)) intersecting the k-sphere 892 of the probe beam k-space distribution876 (E_(p)(

)) contributes to diffraction. This produces the diffracted beam k-spacedistribution 875, (E_(d) (

), constituting the diffracted beam. Because Δn(

) k-space distribution 888 resembles a line segment parallel to thek_(z) axis, it is evident that the magnitude of the angle of reflection882 (θ_(r),) is substantially equal to the magnitude of the angle ofincidence 881 (θ_(i),) so that the hologram exhibits mirror-likebehavior. Furthermore, it is also evident that this property typicallyholds for any incidence angle and wavelength that produces anydiffraction at all, and for any superposition of probe beams producingdiffraction. A k-space polarization distribution Δn(

)*E_(p)(

) will intersect the probe k-sphere at a single point withmirror-symmetry about the k_(x) axis (or about the k_(x), k_(y) plane inthe 3D case). Thus, the hologram of FIG. 8A is configured to exhibitmirror-like behavior at a relatively broad range of wavelengths andangles, and thus constitutes a broadband holographic mirror.

Embodiments typically, but not necessarily, exhibit a gap in Δn(k)k-space distribution 888 near the origin, as shown in FIG. 8A. Thepresence of the gap can limit performance at very high Δθ (i.e., grazingangles of both incidence and reflection).

According to some embodiments, a skew mirror Δn(

) k-space distribution may be rotated to an arbitrary angle with respectto the k_(x), k_(y), and k_(z) axes. In some embodiments, the Δn(

) k-space distribution is not perpendicular to the relevant reflectingsurface in real space. In other words, the reflective axis of a skewmirror embodiment is not constrained to coincident with surface normal.

FIGS. 9A and 9B illustrate a skew mirror in k-space. FIGS. 9A and 9B areidentical to FIGS. 8A and 8B, respectively, excepting that alldistributions and vectors have been rotated by approximately 45° aboutthe origin. Following the discussion of FIG. 8B, it is evident that theskew mirror of FIG. 9B also produces mirror-like diffraction for allprobe beam wavelengths and angles that produce diffraction. Thediffraction is mirror-like with respect to the reflective axis 861defined by the line segment-like Δn(

) k-space distribution 888, e.g., the angle of incidence 881 magnitudewith respect to the reflective axis 861 is equal to the angle ofreflection 882 magnitude with respect to the reflective axis 861. FIG.9B illustrates one such case.

FIG. 10A illustrates the operation of a skew mirror in real space. Skewmirror 1010 is characterized by reflective axis 1038 at angle −13°measured with respect to the z axis, which is normal to the skew mirrorsurface 1012. Skew mirror 1010 is illuminated with incident light 1024with internal incidence angle −26° measured with respect to the z axis.Principal reflected light 1027 is reflected with internal reflectionangle 180° measured with respect to the z axis.

FIG. 10B illustrates the skew mirror 1010 of FIG. 10A in k-space. Linesegment-like Δn(

) k-space distribution 1088 passes through the origin, and has an angleof −13° with respect to the z axis, equal to that of reflective axis1038. Recording k-sphere 1070 is the k-sphere corresponding to thewriting wavelength of 405 nm. A red k-sphere 1090, green k-sphere 1092,and blue k-sphere 1093 in FIGS. 10B and 10D indicate k-spherescorresponding to wavelengths of light residing in the red, green, andblue regions of the visible spectrum, respectively.

FIG. 10C illustrates a highly magnified portion of FIG. 10B showing theleft intersection between recording k-sphere 1070 and line segment-likeΔn(

) k-space distribution 1088 according to an embodiment. In this view,line segment-like Δn(

) k-space distribution 1088 can be seen to be include multiple discreteholograms. Each of the multiple discreet holograms 1005 is representedby a horizontal line demarking the first null-to-first null spacing ofthe hologram in the k_(z) direction. In some embodiments, the spacing ofthe discrete holograms may be higher or lower than illustrated in FIG.10C. In some embodiments, the spacing may be low enough to create gapsin line segment-like Δn(

) k-space distribution 1088. In some embodiments with gaps, the use ofbroadband illumination may substantially mask any effect of the gapsupon the reflected light. In some embodiments, this approach may resultin a net diffraction efficiency increase. In other embodiments, thespacing of the discrete holograms may be so dense as to approximate orbe equivalent to a continuous distribution.

FIG. 10D illustrates the reflection of blue incident light by the skewmirror of FIG. 10A in k-space. Incident light having a probe beam wavevector 1078 impinges with an internal incidence angle of −26° measuredwith respect to the z axis. The tip of probe beam wave vector 1078 lieson blue k-sphere 1093, indicating the position of point-like probe beamk-space distribution 1076 (E_(p)(

)). Polarization density distribution 1080 is given by the convolutionΔn(

)*E_(p) (

), which resembles line segment-like Δn k-space distribution 1088 (seenin FIG. 10C) translated to the tip of probe beam wave vector 1078.Principal reflected light having diffracted beam wave vector 1077 isdetermined from equation (4) by evaluating polarization densitydistribution 1080 at blue k-sphere 1093. Principal reflected lighthaving diffracted beam wave vector 1077 is reflected with internalpropagation angle 180° measured with respect to the z axis.

The term probe beam, sometimes used here when describing skew mirrorproperties in k-space, is analogous to the term incident light, which issometimes used here when describing skew mirror reflective properties inreal space. Similarly, the term diffracted beam, sometimes used herewhen describing skew mirror properties in k-space, is analogous to theterm principal reflected light, sometimes used here when describing skewmirror properties in real space. Thus when describing reflectiveproperties of a skew mirror in real space, it is sometimes stated thatincident light is reflected by a hologram (or other grating structure)as principal reflected light, though to state that a probe beam isdiffracted by the hologram to produce a diffracted beam says essentiallythe same thing. Similarly, when describing reflective properties of askew mirror in k-space, it is sometimes stated that a probe beam isdiffracted by a hologram (or other grating structure) to produce adiffracted beam, though to state that incident light is reflected by thegrating structure to produce principal reflected light has the samemeaning in the context of embodiments of the present disclosure.

As shown in FIG. 10D, probe beam wave vector 1078 and diffracted beamwave vector 1077 necessarily form the legs of a substantially isoscelestriangle with line segment-like polarization density distribution 1080as the base. The equal angles of this triangle are necessarily congruentwith the angle of incidence, 1008, and angle of reflection 1009, bothmeasured with respect to reflective axis 1038. Thus, skew mirror 1010reflects light in a substantially mirror-like manner about reflectiveaxis 1038.

The isosceles triangle construction of FIG. 10D obtains whenever Δn(

) k-space distribution 1088 substantially resembles a segment of a linepassing through the origin, as shown in FIG. 10C. Polarization densitydistribution 1080 hence substantially resembles the straight base of anisosceles triangle, leading to mirror-like reflection about reflectiveaxis 1038 for any incident internal wave vectors of any length thatdiffracts. In some embodiments, dispersion of the grating medium maycause internal wave vectors of the same direction but differing lengthsto refract in different directions in an external medium according toSnell's law. Similarly, dispersion may cause external wave vectors ofthe same direction and differing lengths to refract in differentdirections in the internal grating medium. Accordingly, if it is desiredto minimize the effects of dispersion in a skew mirror, it may bedesirable to impart a curve to line segment-like Δn(

) k-space distribution 1088, or to otherwise deviate from a line thatpasses through the origin. Such an approach may reduce net angulardispersion in reflections involving external refraction according tosome metric. Since the dispersion of useful grating media is typicallyquite low, the required deviation from a straight line passing throughthe origin is small. Adjustments to line segment-like Δn(

) k-space distribution 1088 that compensate for dispersion do not falloutside the scope of the present disclosure.

FIG. 11A illustrates the reflection of green incident light by the skewmirror of FIG. 10A in k-space. Incident light with wave vector 1178Aimpinges with internal propagation angle −35° measured with respect tothe z axis. Principal reflected light with wave vector 1177A isreflected with internal propagation angle −171° measured with respect tothe z axis. The magnitudes of angle of incidence 1108A and angle ofreflection 1109A are both substantially equal to 22 degrees measuredwith respect to reflective axis 1038, thus constituting a mirror-likereflection about reflective axis 1038.

FIG. 11B illustrates the reflection of red incident light by the skewmirror of FIG. 10A in k-space. Incident light having probe beam wavevector 1178B impinges with internal propagation angle −35° measured withrespect to the z axis. Principal reflected light having diffracted beamwave vector 1177B is reflected with internal propagation angle −171°measured with respect to the z axis. The magnitudes of angle ofincidence 1108B and angle of reflection 1109B are both substantiallyequal to 22° measured with respect to reflective axis 1038, thusconstituting a mirror-like reflection about reflective axis 1038.

FIGS. 11A and 11B show the reflection of green and red light at the sameangles of incidence and reflection, illustrating the achromaticreflection property of the skew mirror. Those skilled in the art willrecognize that the geometrical constructions of FIGS. 10A-D and 11A-Bwill produce mirror-like reflection at all angle/wavelength combinationsthat produce reflection, including angles and wavelengths notspecifically illustrated.

These examples are merely illustrative and non-limiting. Embodiments ofa skew mirror effect a mirror-like reflection with respect to internalpropagation angles, external angles must be determined using Snell's lawat the relevant boundaries. Because of this, a skew mirror may introduceaberrations, dispersion, and/or field distortion to external wavefronts.In some embodiments, aberrations, dispersion, and/or field distortionsmay be mitigated by the use of compensating optics. In some embodiments,the compensating optics may include another skew mirror in a symmetricrelationship.

A relatively thin skew mirror may introduce lowered angular resolutionin the reflected beam in proportion to the beam's projection onto thethin axis. In some cases it may be advantageous to increase thethickness of the recording layer in order to mitigate this effect.Embodiments of a skew mirror may be either fully or partiallyreflective. Embodiments of a skew mirror may require relatively highdynamic range recording medium to achieve high reflectivity over arelatively wide wavelength bandwidth and angle range. In an embodiment,a skew mirror with an angular range spanning 105° at 405 nm down to 20°at 650 nm may require 183 individual holograms in a 200 μm recordinglayer. This configuration has a reflectivity of approximately 7.5% usinga photosensitive recording medium with a maximum refractive indexmodulation of 0.03. In some embodiments, increasing recording mediumthickness may not lead to increased reflectivity since diffractiveselectivity also increases with thickness.

The preceding exposition pertains to internal wavelengths andpropagation angles, although in one case a slab-like hologram withthickness in the z direction was described. Many other configurationsare possible within the scope of the invention. Without implyinglimitation, a few exemplary embodiments are illustrated here.

FIG. 12A illustrates an embodiment referred to as a skew windowincluding grating structure 1205 in a grating medium, and including areflective axis 1261 about which incident light is symmetricallyrefracted. The skew window is a transmissive analog of the skew mirror.FIG. 12B shows a skew coupler embodiment, which uses a skew mirror tocouple external light into or out of a waveguide 1294. Transmissive skewcouplers are also possible. FIG. 12C shows a skew prism embodiment,which may fold an optical path and/or invert an image.

FIG. 13A illustrates a pupil relay embodiment formed by a slab waveguide1394 with two skew couplers, each of which comprises a grating medium1310 having a reflective axis 1361 that differs from surface normal ofthe grating medium. Since this device is configured to relay input raysto output rays with a uniform 1:1 mapping, it can transmit an image atinfinity through the waveguide 1394 to the eye or other sensor. Such aconfiguration may be useful for head mounted displays (HMDs), amongother applications. In the reverse direction, it may relay an image ofthe eye, possibly for the purposes of eye tracking. FIG. 13B shows askew mirror 1300 used as a concentrator/diffuser, which can transform alarge dim beam into a bright small one, and/or vice-versa. Theseexamples are merely illustrative. The skew mirrors described herein maybe used in input couplers, output couplers, cross-couplers, interleavedcouplers, diamond expanders, or any other desired light redirectingoptical components in a device or optical system such as a head mounteddisplay device (e.g., a device in which display light is redirected toan eye box, e.g., using waveguides).

FIGS. 14A and 14B illustrate an angle filter embodiment of a skewmirror. In FIG. 14A, a Δn(

) k-space 1488 distribution is indicated with a higher low frequencycut-off (i.e., larger center gap) compared to the distributionillustrated in FIG. 8A. As a consequence, the skew mirror will reflectonly the low θ (i.e., near normal incidence) angular components ofnarrow band incident beam into reflected beam E_(r), while transmittinghigh θ angular components in E_(t). One skilled in the art will readilydiscern that an arbitrary circularly-symmetric transfer function may beso realized by modulating the amplitude and/or phase of the linesegment-like Δn(

) distribution according to an embodiment of the invention. Angularfiltering may also be accomplished with skew mirrors, and inconfigurations involving multiple skew mirrors recorded in one or moremedia. These configurations may not be constrained to becircularly-symmetric, and may achieve some level of achromaticoperation.

FIG. 15 illustrates another skew mirror embodiment that includes severalskew mirrors 1500 whose reflective axes 1561 intersect. A viewer can sitat the point of convergence and see several images of them self. Theseexamples are merely illustrative and non-limiting.

Skew mirrors may be recorded holographically according to an embodiment.Skew mirrors may be recorded holographically or fabricated by withnon-holographic means according to some embodiments. FIGS. 16A and 16Billustrate additional methods for recording skew mirrors. In FIG. 16A,substantially collimated recording beams are used to illuminate agrating medium to create a desired Δn(

) distribution. In one embodiment, illustrated in FIG. 16A, a recordingbeam set consisting of a first recording beam 1654A and a secondrecording beam 1655A at wavelength A illuminate the grating medium 1610in order to record a first point-like subset of the desired linesegment-like Δn(

) distribution, e.g. the highest-frequency components (the outer tips ofΔn(

). The angles of incidence θ₁ and θ₂ of a recording apparatus are thenadjusted to produce another set of recording beams consisting of anotherfirst recording beam 1654B and another second recording beam 1655B,which are also at wavelength λ. The other first and second recordingbeams 1654B, 1655B illuminate the medium to record a second point-likesubset of the desired line segment-like Δn(

) distribution. This process is repeated using yet another set ofrecording beams including yet another first recording beam 1654C and yetanother second recording beam 1655C, etc., until an entire desired linesegment-like Δn(

) distribution has been recorded.

In some embodiments, this recording may be made in one continuousexposure wherein θ_(r) and θ_(s) are adjusted continuously andsynchronously in order to produce the desired distribution. In otherembodiments, separate, discreet exposures where θ_(r) and θ_(s) arefixed during exposure and changed only between exposures are used. Stillother embodiments may combine these methods. In some embodiments, thecomponents of Δn(

) may be written in an arbitrary order. In some embodiments, intensitymay be varied across one or both beams in order to control the spatialdiffraction efficiency profile. In some embodiments, a phase controlelement (e.g., a mirror mounted on a piezo-electric actuator) may beinserted into one or both beam paths in order to control the phase ofeach exposure. In some embodiments, more than one skew mirror orbroadband skew mirror might be recorded into the same medium.

In the case of discrete exposures, the number and angular density ofexposures is sufficient to produce a smooth, continuous linesegment-like Δn(

) distribution. In some embodiments, exposures are made at angularincrements corresponding to a function of this angular selectivity,e.g., at the angular spacing of the full-width-quarter-maximum (FWQM) ofthe diffraction efficiency peaks. In other embodiments, the angularexposure density might be finer than this in order to assure a smoothfinal distribution.

The number of FWQM peaks necessary to span the line segment-like Δn(

) distribution may be regarded as an equivalent number of holograms, M,required to form the distribution. Accordingly, the maximum possiblediffraction efficiency of the resulting skew mirror may be estimated byη=(M/M/#)² where η is the diffraction efficiency, and M/# is a materialparameter characterizing the dynamic range of the recording medium. Thisestimate may be refined according to the geometry of each individualexposure or the overlap of neighboring exposures.

FIG. 16B illustrates an embodiment where a first prism 1659A and asecond prism 1659B are incorporated to produce internal beam angles thatare not otherwise accessible due to refraction at the grating medium1610 surface. This method is typically used, for example, to fabricatethe skew coupler of FIG. 12B. Configurations of FIGS. 13A and 13B mayalso achieve a desired distribution.

In some embodiments, a single recording wavelength λ may be chosen towrite the entire line segment-like Δn(

) distribution. For example, in an embodiment it is possible to write askew mirror that operates across all visible wavelengths using only a405 nm laser source. This has an advantage of requiring sufficientrecording medium sensitivity at only a single wavelength, as well as anadvantage of simplicity. In some embodiments, more than one recordingwavelength is used. In still other cases, a continuously-variablewavelength source is used. In one such embodiment, the recording anglesθ_(r) and θ_(s) are held constant, and the recording wavelength isinstead changed in order to produce the entire line segment-like Δn(

) distribution, or a subset thereof.

Other methods for producing a skew mirror fall within the scope of thepresent disclosure. In some embodiments, for example, a very thickdielectric layer structure is built up using optical coating means. Thestructure is designed to produce broadband reflectivity withinsub-layers, typically by repetition of a conventional broadbandreflective coating design. The thick structure is then ground andpolished to produce a surface at an oblique angle to the coating layers.The resulting structure typically exhibits mirror-like behavior withrespect to a reflective axis substantially defined by the normal of thecoating layers rather than the polished surface, and thus constitutes askew mirror. In some embodiments, atomically-precise manufacturingmethods enable fabrication of skew mirrors by composing dielectricstructures atom-by-atom without regard to external surfaces.

Skew mirrors may be said to be non-flat in two senses: 1) When thephysical shape of the recording medium is not flat; and 2) when theholographic fringes are not planar. Embodiments of mirrors according tothe present disclosure, including examples of skew mirrors, broadbandmirrors, and holographic mirrors, include holograms recorded in mediumthat is not slab-like in shape. In an example, in an embodiment, arecording layer is cast with a uniform thickness, but on a curvedsurface. In another example, a non-uniform recording layer (e.g.,wedge-shaped) is utilized. In still another example, an arbitrary shape(e.g., spherical) is molded. In these non-slab-like mirror cases,whether the designation “skew mirror” is appropriate depends on thegeometry of the relevant surface(s). Non-slab-like holographic mirrorstypically exhibit broadband mirror-like properties.

In some embodiments, it is desirable to introduce optical power or otherdeliberate aberrations into a reflection. This can be accomplished withembodiments of a skew mirror by locally varying the direction of thereflective axis, for example so that a plane-wave incident beam isreflected to form a spherical-wave reflected beam, as occurs with aconventional parabolic mirror. Such a skew mirror can be fabricated, forinstance, by using one converging and one diverging beam in thefabrication method of FIG. 13 and by recording while changing thewavelength instead of the angle of incidence. Such a mirror can also befabricated by polishing dielectric layers deposited on a non-flatsurface, or by using advanced atomically-precise manufacturing methods.

Some holographic recording system embodiments incorporates mirrors,lenses and prisms to direct first and second recording beams into thegrating medium in such a way that translation of the grating medium isnot required to record multiple holograms at varying recording beaminternal angles, at approximately the same location in the gratingmedium.

In some embodiments a prism in addition to the coupling prism may beused to fabricate the skew mirror. In some embodiments a variety ofcoupling prisms and flat pieces of glass may be used. In someembodiments multiple beams, E_(r_N) and E_(s_N), at multiplewavelengths, λ_(N), may be. In some embodiments multiple wavelengths maybe used to fabricate multiple discrete line segment-like Δn(

) distributions. In some embodiments multiple wavelengths may be used tofabricate a line segment-like Δn(

) distribution that may be continuous or may include closely spacedsections. In some embodiments the incident angle of the signal and/orreference beam may be adjusted to compensate for shrinkage of the samplematerial. In some embodiments the sample may be rotated to compensatefor shrinkage of the sample material. In some embodiments the wavelengthmay be changed to compensate for shrinkage of the sample material.

Two holograms may be said to be pair-wise coherent if illumination by aprobe beam produces diffracted beams substantially in a desired fixedphase relationship. In the case of non-orthogonal probe or diffractedbeams, pair-wise coherence may manifest as a constructive or destructiveinterference pattern between diffracted beams. When input light at oneinput angle diffracts off of two holograms, the diffracted light may become out at slightly different angles (e.g., within the angularresolution of the skew mirror). Conversely, light at two slightlydifferent input angles (e.g., within the angular resolution of the skewmirror) may diffract off of the two holograms to merge into the sameangle out.

Of particular interest is the case of neighboring holograms within askew mirror. The neighboring holograms may, for example, be adjacentholograms in the skew mirror. The magnitude of a difference in gratingfrequency between any two holograms in a skew mirror described hereinmay sometimes be referred to as frequency gap |ΔK_(G)|. Frequency gap|ΔK_(G)| can be a useful metric for describing hologram “spacing” (e.g.how close to each other in k-space the grating vectors for the any twoholograms are). The frequency gap |ΔK_(G)| between a given hologram andan adjacent hologram (e.g., in k-space) may sometimes be referred to asthe adjacent frequency gap |ΔK_(G)|.

Among a set of multiple holograms (e.g., a set of volume holographicgratings in a skew mirror), each hologram in the set has a correspondinggrating vector in k-space. The grating vector has a correspondinggrating vector magnitude K_(G). A first hologram in the set is sometimesreferred to as being “adjacent” or “neighboring” to a second hologram inthe set of holograms when the second hologram has the next highest ornext lowest grating vector magnitude K_(G) relative to the gratingvector magnitude of the first hologram (among the holograms in the set).Each hologram in the set may be separated from one or two adjacentholograms in the set by an adjacent frequency gap |ΔK_(G)|. The adjacentfrequency gap |ΔK_(G)| may be the magnitude of the difference betweenthe grating vector magnitudes K_(G) for the adjacent holograms. Forexample, the first hologram in the set may have a first grating vectormagnitude K_(G1), the second hologram in the set may have a secondgrating vector magnitude K_(G2), and the first grating vector magnitudeK_(G1) may be separated from the second grating vector magnitude K_(G2)in k-space by the adjacent frequency gap |ΔK_(G)| (e.g., the first andsecond holograms may be “adjacent” or “neighboring” holograms in theskew mirror).

Each hologram in the set (skew mirror) is separated from one or moreother holograms in the set by a corresponding adjacent frequency gap|ΔK_(G)| (e.g., the adjacent frequency gaps across the set need not beuniform). In some embodiments, the mean adjacent frequency gap |ΔK_(G)|for the entire set of holograms may influence the performance of theskew mirror. The grating vector magnitude K_(G) of a given hologram maydetermine the grating frequency for the hologram (e.g., the frequency ofrefractive index modulations in the grating medium in physical space aswell as the wavelength of light that is Bragg matched to the hologram).Grating vector magnitude K_(G) may therefore sometimes be referred toherein as grating frequency K_(G). Each hologram in the set of hologramshas a corresponding grating frequency K_(G). The direction of thegrating vector associated with grating frequency K_(G) may give thedirection (orientation) of the refractive index modulations in thegrating medium in physical space. Grating frequency K_(G) and thefrequency gap |AK_(G)| may be expressed in various units, including, butnot limited to, radians per meter (rad/m) and/or sinc peak to sincnulls.

A relatively small mean adjacent frequency gap |ΔK_(G)| for the set ofholograms can correspond to relatively high skew mirror image fidelity(e.g., for the entire set of holograms). However, a relatively smallmean adjacent frequency gap |ΔK_(G)| may result in an overlap betweenthe diffracted beams of neighboring holograms, allowing them tointerfere with each other. Also, where the mean adjacent frequency gap|ΔK_(G)| for a set of holograms is relatively small, the total number ofholograms in the set is larger in order to span a given adjacentfrequency gap |ΔK_(G)| range for the set. Moreover, given that recordingcapacity for grating mediums is typically limited by dynamic range(usually expressed as Δn), recording more holograms in a set usuallymeans that each hologram in the set is weaker (e.g., is recorded morefaintly in the medium). Accordingly, tension exists between havingrelatively small adjacent frequency gaps |ΔK_(G)| for a set of holograms(which requires more holograms, other things being equal), and havinglarger adjacent frequency gaps |ΔK_(G)| for the set, which allowsrecording of fewer, but stronger holograms.

Neighboring holograms within a skew mirror are of particular interestwith respect to pair-wise coherence, because neighboring holograms mayproduce non-orthogonal overlapping diffracted beams in response to asingle plane-wave probe beam component. In these situations, it may bedesirable to control the pair-wise coherence of the holograms in orderto achieve constructive interference among overlapping diffractive beams(sometimes referred to herein as “constructive coherence”). For example,this may prevent dark lines from appearing in reflected images usingnarrow band illumination due to destructive interference betweenneighboring holograms in the skew mirror. A skew mirror withconstructive pair-wise coherence and dense holograms may closelyapproximate the performance of an idealized skew mirror written withscanning beams in a single exposure, rather than a series of exposuresas used to manufacture skew mirrors in practice. In other embodiments,it may be desirable to deliberately create destructive interference orsome other interference state. In some embodiments, a 90 degree phasedifference between overlapping diffracted beams may be desirable so thatneither constructive nor destructive interference will manifest.

FIG. 17 is a diagram showing how a skew mirror may include pair-wisecoherent gratings (holograms). As shown in FIG. 17, a skew mirror 1700(e.g., a set of gratings/holograms) may be formed in grating medium1702. The skew mirror may be used for any of the optical devicesdescribed above and may exhibit diffractive properties as describedabove in connection with FIGS. 1-16, for example. Skew mirror 1700 mayinclude a set of at least partially overlapping holograms within a givenvolume of grating medium 1702. Each hologram in the set may have adifferent respective grating frequency (e.g., grating vectors withdifferent magnitudes but oriented in the same direction). Each hologrammay be configured to redirect incident light into a given direction suchas direction 1704 (e.g., towards an eye-box in scenarios where the skewmirror is used to form an output coupler).

At least two of the holograms in the set (e.g., in skew mirror 1700) maybe pair-wise coherent. As an example, the at least two holograms thatare pair-wise coherent may be adjacent holograms in the set (skewmirror). The at least two holograms are pair-wise coherent ifillumination by a probe beam produces diffracted beams substantially ina desired fixed phase relationship (e.g., if input light to eachhologram is coherent and the corresponding light diffracted by eachhologram is also coherent). For example, the at least two holograms mayinclude a first hologram having a first grating frequency K_(G1) and asecond hologram having a second grating frequency K_(G2) that isdifferent from the first grating frequency (e.g., adjacent to the firstgrating frequency in k-space). The first and second holograms may bothreceive coherent probe beam 1706 from a light source or other opticalcomponents. The first hologram may diffract probe beam 1706, as shown bydiffracted beam 1708. The second hologram may also diffract probe beam1710, as shown by diffracted beam 1710. Diffracted beams 1708 and 1710may be coherent (e.g., in a desired fixed phase relationship with eachother). In this way, the first and second holograms may be pair-wisecoherent, for example. This example is merely illustrative. In general,any desired number of holograms in skew mirror 1700 may be pair-wisecoherent (e.g., may each diffract light that is coherent with thediffracted light from the other holograms in response to a coherentprobe beam). In one practical example, probe light 1706 may correspondto light of a single frequency that has been emitted from a singledisplay pixel spanning a very small range of angles (e.g., 0.02 degreesor less). Diffracted beams 1708 and 1710 may correspond to overlappingimages or partial images of this display pixel. Depending on the phaserelation between diffracted beams 1708 and 1710, the pixel may appear ofuniform illumination, or may exhibit a bright or dark band caused byinterference between diffracted beams 1708 and 1710.

It may sometimes be desirable for the peak in the diffraction responsefor a first hologram to lie within a null of the diffraction responsefor a pair-wise coherent and adjacent second hologram in the skewmirror. It may be especially important for the first and secondholograms to be pair-wise coherent in this example. FIG. 18 shows adiagram of diffracted light as a function of wavelength for theseillustrative adjacent first and second holograms. As shown in FIG. 18,the first hologram may have a first grating frequency (e.g., K_(G1))that configures the first hologram to exhibit a diffraction response asshown by curve 1800. Similarly, the second hologram may have a secondgrating frequency (e.g., K_(G2) separated from the first gratingfrequency by an adjacent frequency gap) that configures the secondhologram to exhibit a diffraction response as shown by curve 1802.

As shown by curve 1802, the diffraction response of the second hologramis configured so that the spectral peak 1806 in the diffraction responseof the second hologram lies within a spectral null 1810 of thediffraction response of the first hologram. Similarly, the diffractionresponse of the first hologram is configured so that spectral peak 1804in the diffraction response of the first hologram lies within a spectralnull 1808 in the diffraction response of the second hologram. Thediffraction peaks may, if desired, also be configured so that thespectral peaks of side-lobes 1812 for one hologram align with thespectral nulls of the side-lobes 1812 for the other hologram. Whenconfigured in this way, an incident photon at wavelength 1814 willdiffract off of either the first or second hologram. If the first andsecond holograms are not pair-wise coherent, this may produce a blackline in the diffracted light when viewed at a screen that is displacedfrom the holograms. Configuring the first and second holograms to bepair-wise coherent may serve to mitigate these issues.

In practice, it can be difficult to manufacture skew mirrors havingpair-wise coherent holograms. Manufacturing (writing) apparatuses forthese types of skew mirrors may need to be very precise in order toensure that two or more holograms in the skew mirror are pair-wisecoherent. Consider, for example, a scenario where a writing apparatus ofthe type shown in FIG. 3 is used.

In theory, a skew mirror writer could achieve constructive coherence byexactly path-length matching the recording beams (e.g., beams 354 and355 in FIG. 3) at the same location within the recording medium over allbeam angles and stage positions. In some embodiments, smallmisalignments and errors in stage motion may introduce phase errors inthe written holograms, thus producing non-pair-wise coherence inadjacent holograms of the skew mirror.

In some embodiments, passive alignment of a skew mirror writer allowsthe holograms to exhibit pair-wise coherence. FIG. 19 is a diagramshowing an example of one possible manufacturing system for producingcoherent skew mirrors with pair-wise coherent holograms. As shown inFIG. 19, system 1900 (sometimes referred to herein as manufacturingsystem 1900, manufacturing apparatus 1900, writing system 1900, writingapparatus 1900, recording system 1900, or recording apparatus 1900) maybe provided for writing coherent skew mirrors with pair-wise coherentholograms.

In system 1900 of FIG. 19, the prism package containing the recordingmedium of FIG. 3 has been replaced with a non-polarizing beam splitter1902 substantially aligned with the axis of translation 1906 (sometimesreferred to herein as stage axis 1906 or translation axis 1906) forstage carrier 1904. Mirror 1908 may reflect signal beam 1910 towardsbeam splitter 1902. Mirror 1912 may reflect reference beam 1914 towardsbeam splitter 1902. Mirror 1908 may be rotated to provide signal beam1910 at angle θ_(S). Mirror 1912 may be rotated to provide referencebeam 1914 at angle θ_(R). Stage carrier 1904 may be translated alongstage axis 1906. Beam splitter 1902 may be rotated by angle θ_(B) withrespect to stage axis 1906.

Signal beam 1910 may reflect off of beam splitter 1902 towards detector1916. Detector 1916 may be a photodetector (image detector) or othersensor that senses light. In another suitable arrangement, detector 1916may be a card, wall, or screen for viewing light that hits the card,wall, or screen. Reference beam 1914 may transmit through beam splitter1902 towards detector 1916. This is merely illustrative and, if desired,the reference beam may reflect off of the beam splitter whereas thesignal beam is transmitted by the beam splitter. This may permit thereflected signal beam 1910 to be combined with the transmitted referencebeam 1914 (or vice-versa), forming an interference pattern 1918 atdetector 1916 indicative of the relative phases between beams 1910 and1914. Two holograms will exhibit constructive pair-wise coherence whenthe fringes in pattern 1918 are in the same phase at both writingconditions (e.g., the fringes appear to remain stationary when changingfrom one condition to another).

In order to achieve this condition, system 1900 may be aligned using analignment method such as the following. First, angle θ_(S) may be set toa particular value. Next, angle θ_(R) may be adjusted to generate coarsefringes in pattern 1918. The position of stage carrier 1904 maysubsequently be adjusted (e.g., via translation along axis 1906). Next,beam splitter 1902 may be rotated to adjust angle θ_(B) until fringemotion is reduced in pattern 1918. This may serve to align fringe planeswith stage axis 1906 and, in some arrangements, to align the surface ofbeam splitter 1902 to the stage axis (e.g., so that angle OB isapproximately zero). Once this is achieved, the corresponding pair ofangles θ_(S) and θ_(R) may be saved and used to record pair-wisecoherent holograms on a grating medium placed on stage carrier 1904.This alignment method is merely illustrative and, in general, otheralignment strategies may be employed to achieve the fine alignmentnecessary to write pair-wise coherent holograms in a skew mirror.

In some embodiments, an active alignment method may be used to activelyalign the writing system while the pair-wise coherent holograms arebeing written (or before/between writings of multiple holograms). Suchan active alignment method may improve the pair-wise coherence relativeto the initial alignment method described above in connection with FIG.19. FIG. 20 is a diagram showing an example of one possiblemanufacturing system for producing coherent skew mirrors with pair-wisecoherent holograms using active alignment methods.

As shown in FIG. 20, system 2000 (sometimes referred to herein asmanufacturing system 2000, manufacturing apparatus 2000, writing system2000, writing apparatus 2000, recording system 2000, or recordingapparatus 2000) may be provided for writing coherent skew mirrors withpair-wise coherent holograms using active alignment methods. As shown inFIG. 20, system 2000 may include prisms 2002 and 2004 on opposing sidesof grating medium 2006, on which pair-wise coherent holograms in a skewmirror are to be written. Signal beam 2016 may be provided by a firstlight source (e.g., via a reflecting mirror, etc.). Reference beam 2020may be provided by a second light source (e.g., via a reflecting mirror,etc.).

Active alignment structures may be mounted over or under prisms 2002 and2004 (e.g., on top of the prisms or otherwise in a plane above or belowthe prisms). For the sake of illustration, the active alignmentstructures are described herein as being mounted over prisms 2002 and2004. The active alignment structures may include partial reflector 2008(e.g., a partial reflector such as a 50% partial reflector) and detector2010 (e.g., a linear detector, image sensor, or other light sensor).Partial reflector 2008 may be stationary or may rotate (e.g., for fieldof view expansion). Light redirecting components such as periscopes 2012and 2014 may extend from the plane of prisms 2002 and 2004 upwards(e.g., parallel to the Z-axis) to the plane of partial reflector 2008and detector 2010. Periscope 2012 may redirect a portion of signal beam2016 upwards into the plane of partial reflector 2008 and detector 2010,as shown by beam 2018. Beam 2018 may reflect off of partial reflector2008 towards detector 2010. Similarly, periscope 2014 may redirect aportion of reference beam 2020 upwards into the plane of partialreflector 2008 and detector 2010, as shown by beam 2026. Beam 2026 maypass through partial reflector 2008 and combine with signal beam 2018 toproduce a combined beam 2028 that is received by detector 2010. Theexample of FIG. 20 is merely illustrative and, in another suitablearrangement, signal beam 2018 may transmit through partial reflector2008 whereas reference beam 204 reflects off of partial reflector 2008.

While periscope 2012 redirects some of signal beam 2016 towards partialreflector 2008, periscope 2012 may allow the remainder of signal beam2016 to pass to grating medium 2006 (e.g., in the plane below the activealignment structures), as shown by beam 2020. Similarly, periscope 2014may allow the remainder of reference beam 2020 to pass to grating medium2006, as shown by beam 2024. Beams 2020 and 2024 may be used to writepair-wise coherent holograms for a corresponding skew mirror in gratingmedium 2006. Detector 2010 may measure fringe patterns associated withsignal beam 2016 and reference beam 2020. Detector 2010 may providesensor data (e.g., image data or other data identifying the intensity ofcombined beam 2028 across its length and thus the interference/fringepattern produced by combining beams 2018 and 2026) to controller 2034over control path 2036. Controller 2034 may process this data todetermine whether signal beam 2016 is suitably aligned with referencebeam 2020 for recording pair-wise coherent holograms (e.g., byidentifying and analyzing fringe patterns in the data).

If controller 2034 determines that the signal beam and reference beamare not suitably coherent (aligned) based on the data received fromdetector 2010, controller 2034 may control alignment adjustmentstructure 2030 over path 2040 or alignment adjustment structure 2032over path 2038 to adjust the physical path lengths or effective pathlengths (relative phases) of signal beams 2016 and 2020 (e.g., to adjusta difference in physical or effective path length between the beams)until the beams are suitably coherent. Alignment adjustment structures2030 and 2032 may include, for example, piezo-electric actuators,translational structures, rotational structures, piezo-actuated phaseshifting mirrors, and/or any other desired structures for adjusting theeffective or physical path lengths of signal beam 2016 and referencebeam 2020. This process may be performed iteratively (e.g., using activefeedback from detector 2010) until the beams are suitably adjusted forrecording pair-wise coherent holograms on medium 2006.

Masking structures (not shown) may be used to shield grating medium 2006from the signal and reference beams during this active alignmentprocedure if desired. For example, these active alignment procedures maybe performed immediately prior to exposure of each hologram in gratingmedium 2006 (e.g., so that measurements and adjustments may be madeimmediately prior to the exposure of each hologram in order to assurepair-wise coherence with the preceding hologram). In general,measurement with detector 2010 and/or adjustment using structures 2030and/or 2032 may be performed at any desired times, including duringexposure of a hologram on grating medium 2006. In this example, thewriting beams themselves are used for measurement and alignment. This ismerely illustrative and, in another suitable arrangement, separate testbeams may be used to perform measurement and alignment.

FIG. 21 shows a perspective view of an exemplary periscope 2104 that maybe used in forming periscope 2012 and/or periscope 2014 of FIG. 21. Asshown in FIG. 21, periscope 2104 may include a first slanted/reflectivesurface 2106 that is interposed in the aperture associated with beam2102 (e.g., signal beam 2016 or reference beam 2020 of FIG. 20). Surface2106 may reflect a portion of beam 2102 upwards (e.g., parallel to theZ-axis) to slanted/reflective surface 2110. Surface 2110 may reflectthis portion of beam 2102 outwards as beam 2108 (e.g., towards partialreflector 2008 of FIG. 20). The remainder of beam 2102 may continue topropagate, as shown by beam 2112 (e.g., towards the grating medium). Asshown in FIG. 21, beam 2108 may extend parallel to beam 2112 but in aplane over beam 2112. This may allow the same writing light to both beused for recording holograms and for sampling to measure and align thewriting system. The example of FIG. 21 is merely illustrative. Ifdesired, surface 2110 may be separated from surface 2106 by any desireddistance (e.g., so that beam 2108 is located within a desired plane suchas the plane of partial reflector 2008 of FIG. 20).

FIG. 22 shows a perspective view of an exemplary periscope 2200 that maybe used in forming periscope 2012 and/or periscope 2014 of FIG. 21. Asshown in FIG. 22, periscope 2200 may include relatively wide slanted(reflective) surfaces 2204 and 2202 for redirecting a portion of arelatively wide signal beam 2206 towards the beam splitter, as shown byarrow 2208. This may, for example, allow a sample slit to travel alongdifferent portions of the width of the beam. In this way, the periscopesin the writing system may raise the beams of light so that the light canbe measured and the system can be adjusted to provide sufficientlycoherent reference and signal beams for writing pair-wise coherentholograms in the skew mirror. The examples of FIGS. 20-22 are merelyillustrative. The periscopes shown in FIGS. 20-22 may be replaced withany desired light redirecting elements.

FIG. 23 is a flow chart of illustrative steps that may be used inperforming active alignment for writing coherent skew mirrors withpair-wise coherent holograms (e.g., using system 2000 of FIG. 20 orusing any other desired writing apparatus). At step 2300, the initialsettings of the writing system (e.g., system 2000 of FIG. 20) may beset. The initial settings may involve setting the signal and referencebeam angles, path lengths, and/or the prism and grating medium positionand orientation relative to the beams (e.g., for recording a desiredhologram in the grating medium).

At step 2302, the periscopes (e.g., periscopes 2012 and 2014 of FIG. 20)may redirect a portion of the signal and reference beams to the beamsplitter (e.g., beam splitter 20008 of FIG. 20), which combines thebeams and provides the combined beam to the detector (e.g.,photodetector 2010 of FIG. 20). The detector may sample the combinedbeam and provide corresponding signals to the controller.

At step 2304, the controller (e.g., controller 2034 of FIG. 20) mayprocess the signals generated by the detector. For example, thecontroller may measure grating phases being recorded based on fringes inthe signals generated by the detector. The controller may determinewhether the signal and reference beams are sufficiently coherent (e.g.,sufficiently coherent to record pair-wise coherent holograms in thegrating medium). If the beams are sufficiently coherent, processing mayloop back to step 2302 as shown by path 2306 to continue to monitor thealignment of the system (or the controller may wait until the nexthologram is to be written, in which case processing will return to step2300 to set the system up for writing the next hologram). If the beamsare not sufficiently coherent (e.g., such that pair-wise coherentholograms will not be written in the grating medium), processing mayproceed to step 2310, as shown by path 2308.

At step 2312, the controller may adjust one or both of the beam pathlengths or phases (effective path lengths) using suitable adjustmentstructures (e.g., structures 2030 and/or 2032 of FIG. 20). As anexample, the controller may adjust a piezo-actuated phase shiftingmirror that passes the signal beam to adjust a phase (effective pathlength) of the signal beam. Processing may subsequently loop back tostep 2302, as shown by path 2312, to sample whether this adjustment hasproduced sufficiently aligned beams. This process may be repeated(iterated) until coherent beams are provided that are suitable forwriting pair-wise coherent holograms.

The steps of FIG. 23 may, for example, be repeated for each hologram tobe written. The steps of FIG. 23 may be performed before a gratingmedium has been placed in the writing apparatus or after a gratingmedium has been placed in the writing apparatus. The steps of FIG. 23may be performed prior to each hologram exposure and/or during hologramexposure. The steps of FIG. 23 may be performed using the writing beamsor using separate test (probe) beams. For example, a first probe beamextending parallel to the signal beam and a second probe beam extendingparallel to the reference beam may be directed towards the partialreflector shown in FIG. 20 instead of the signal and reference beamsused for writing the holograms. The probe beams trace optical pathsparallel to these writing beams. For example, the probe beams mayreflect off of mirrors 152A and 152B of FIG. 4 (or separate mirrorsstiffly coupled to mirrors 152A and 152B), the partial reflector may bemounted to the same structure as medium 110 of FIG. 4, and the probebeams may reflect off of any upstream mirrors reflecting the writingbeams where path length perturbations may be introduced. This example ismerely illustrative and, in general, other operations may be performedto write pair-wise coherent holograms for the skew mirror, desired. Thesteps of FIG. 23 and the other methods and operations described hereinmay be performed by executing instructions (e.g., computer code) storedon storage circuitry such as memory using a processor (e.g., amicroprocessor, CPU, or other processing circuitry at controller 2034 ofFIG. 20). Non-transitory, computer-readable storage media may be used tostore the executed code for performing these steps, for example.

In some embodiments, the system of FIG. 20 may be configured to enablethe recording of pair-wise coherent holograms in the presence ofmechanical disturbances such as vibration (e.g., physical vibrationsfrom nearby or distant footsteps, passing vehicles, seismic activity,sound, weather conditions, etc.). For example, controller 2034 of FIG.20 may include a servo controller that uses feedback from sensor 2010 tocontrol (actuate) alignment adjustment structures 2030 and/or 2032 sothat the coherent phase condition is dynamically maintained regardlessof the presence of vibration or other motion. In some embodiments, sucha servo may allow the system to record pair-wise coherent holograms morequickly by reducing the settling time required after mechanical stagemotions are performed. In some embodiments, such a servo may allow thesystem to record pair-wise coherent holograms while mechanical stagemotions are in progress.

While various embodiments have been described and illustrated herein,other means and/or structures for performing the function and/orobtaining the results and/or one or more of the advantages describedherein may be used, and each of such variations and/or modifications isdeemed to be within the scope of the embodiments described herein. Moregenerally, all parameters, dimensions, materials, and configurationsdescribed herein are merely illustrative and actual parameters,dimensions, materials, and/or configurations may depend upon thespecific application or applications for which the embodiments is/areused. The embodiments may be practiced in any desired combination. Also,various concepts may be embodied as one or more methods, devices orsystems, of which an example has been provided. The acts performed aspart of a method or operation may be ordered in any suitable way.Accordingly, embodiments may be constructed in which acts are performedin an order different than illustrated, which may include performingsome acts simultaneously, even though shown as sequential acts inembodiments. As used herein, the phrase “at least one,” in reference toa list of one or more elements, should be understood to mean at leastone element selected from any one or more of the elements in the list ofelements, but not necessarily including at least one of each and everyelement specifically listed within the list of elements and notexcluding any combinations of elements in the list of elements.Transitional phrases such as “comprising,” “including,” “carrying,”“having,” “containing,” “involving,” “holding,” “composed of,” and thelike are to be understood to be open-ended, i.e., to mean including butnot limited to. The term “approximately,” refers to plus or minus 10% ofthe value given.

The term “approximately” as used herein refers to plus or minus 10% ofthe value given. The term “about,” refers to plus or minus 20% of thevalue given. The term “principally” with respect to reflected light,refers to light reflected by a grating structure. Light that isprincipally reflected at a recited angle includes more light than isreflected at any other angle (excluding surface reflections). Light thatis principally reflected about a recited reflective axis includes morereflected light than is reflected about any other reflective axis(excluding surface reflections). Light reflected by a device surface isnot included when considering principally reflected light. The term“reflective axis” refers to an axis that bisects an angle of incidentlight relative to its reflection. The absolute value of an angle ofincidence of the incident light relative to the reflective axis is equalto the absolute value of the angle of reflection of the incident light'sreflection, relative to the reflective axis. For conventional mirrors,the reflective axis is coincident with surface normal (i.e., thereflective axis is perpendicular to the mirror surface). Conversely,implementations of skew mirrors according to the present disclosure mayhave a reflective axis that differs from surface normal, or in somecases may have a reflective axis that is coincident with surface normal.A reflective axis angle may be determined by adding an angle ofincidence to its respective angle of reflection, and dividing theresulting sum by two. Angles of incidence and angles of reflection canbe determined empirically, with multiple measurements (generally threeor more) used to generate a mean value.

The term “reflection” and similar terms are used in this disclosure insome cases where “diffraction” might ordinarily be considered anappropriate term. This use of “reflection” is consistent withmirror-like properties exhibited by skew mirrors and helps avoidpotentially confusing terminology. For example, where a gratingstructure is said to be configured to “reflect” incident light, aconventional artisan might prefer to say the grating structure isconfigured to “diffract” incident light, since grating structures aregenerally thought to act on light by diffraction. However, such use ofthe term “diffract” would result in expressions such as “incident lightis diffracted about substantially constant reflective axes,” which couldbe confusing. Accordingly, where incident light is said to be“reflected” by a grating structure, persons of ordinary skill in art,given the benefit of this disclosure, will recognize that the gratingstructure is in fact “reflecting” the light by a diffractive mechanism.Such use of “reflect” is not without precedent in optics, asconventional mirrors are generally said to “reflect” light despite thepredominant role diffraction plays in such reflection. Artisans ofordinary skill thus recognize that most “reflection” includescharacteristics of diffraction, and “reflection” by a skew mirror orcomponents thereof also includes diffraction.

The term “light” refers to electromagnetic radiation. Unless referenceis made to a specific wavelength or range of wavelengths, such as“visible light,” which refers to a part of the electromagnetic spectrumvisible to the human eye, the electromagnetic radiation can have anywavelength. The terms “hologram” and “holographic grating” refer to arecording of an interference pattern generated by interference betweenmultiple intersecting light beams. In some examples, a hologram orholographic grating may be generated by interference between multipleintersecting light beams where each of the multiple intersecting lightbeams remains invariant for an exposure time. In other examples, ahologram or holographic grating may be generated by interference betweenmultiple intersecting light beams where an angle of incidence of atleast one of the multiple intersecting light beams upon the gratingmedium is varied while the hologram is being recorded, and/or wherewavelengths are varied while the hologram is being recorded (e.g., acomplex hologram or complex holographic grating).

The term “sinusoidal volume grating” refers to an optical componentwhich has an optical property, such as refractive index, modulated witha substantially sinusoidal profile throughout a volumetric region. Each(simple/sinusoidal) grating corresponds to a single conjugate vectorpair in k-space (or a substantially point-like conjugate pairdistribution in k-space). The term “diffraction efficiency” refers tothe ratio of the power of reflected light to incident light and on agrating medium. The term “entrance pupil” refers to a real or virtualaperture passing a beam of light, at its minimum size, entering intoimaging optics. The term “eye box” refers to a two-dimensional areaoutlining a region wherein a human pupil may be placed for viewing thefull field of view at a fixed distance from a grating structure. Theterm “eye relief” refers to a fixed distance between a grating structureand a corresponding eye box. The term “exit pupil” refers to a real orvirtual aperture passing a beam of light, at its minimum size, emergingfrom imaging optics. In use, the imaging optics system is typicallyconfigured to direct the beam of light toward image capture means.Examples of image capture means include, but are not limited to, auser's eye, a camera, or other photodetector. In some cases, an exitpupil may comprise a subset of a beam of light emerging from imagingoptics.

The term “grating medium” refers to a physical medium that is configuredwith a grating structure for reflecting light. A grating medium mayinclude multiple grating structures. The term “grating structure” refersto one or more gratings configured to reflect light. In some examples, agrating structure may include a set of gratings that share at least onecommon attribute or characteristic (e.g., a same wavelength of light towhich each of the set of gratings is responsive). In someimplementations, a grating structure may include one or more holograms.In other implementations, a grating structure may include one or moresinusoidal volume gratings. In some examples, the grating structures maybe uniform with respect to a reflective axis for each of the one or moregratings (e.g., holograms or sinusoidal gratings). Alternatively oradditionally, the grating structures may be uniform with respect to alength or volume for each of the one or more gratings (e.g., hologramsor sinusoidal volume gratings) within the grating medium. Skew mirrorsas described herein may sometimes also be referred to herein as gratingstructures, holographic grating structures, or volume holographicgrating structures.

The above-described embodiments can be implemented in any of numerousways. For example, embodiments of designing and making the technologydisclosed herein may be implemented using hardware, software or acombination thereof. When implemented in software, the software code canbe executed on any suitable processor or collection of processors,whether provided in a single computer or distributed among multiplecomputers.

In accordance with an embodiment, an optical device is provided thatincludes a waveguide having first and second waveguide substrates, agrating medium between the first and second waveguide substrates, aprism mounted to the first waveguide substrate, where the prism isconfigured to couple light into the grating medium, the grating mediumhas a first abbe number, and the prism has a second abbe number that isdifferent from the first abbe number, and a set of overlapping hologramsin the grating medium, where the set of overlapping holograms isconfigured to direct a first wavelength of the light coupled into thegrating medium in a given direction through a given one of the first andsecond waveguide substrates and is configured to direct a secondwavelength of the light coupled into the grating medium in the givendirection through the given one of the first and second waveguidesubstrates.

The foregoing is merely illustrative and various modifications can bemade to the described embodiments. The foregoing embodiments may beimplemented individually or in any combination.

What is claimed is:
 1. A display system configured to direct images toan eye box, the display system comprising: a waveguide; a grating mediumon the waveguide and configured to receive light that includes at leasta portion of the images; a first hologram in the grating medium, whereinthe first hologram has a first grating frequency and is configured todiffract the light as first diffracted light having a first spectralpeak and a spectral null; and a second hologram in the grating medium,wherein the second hologram has a second grating frequency that isdifferent from the first grating frequency, the second hologram isconfigured to diffract the light as second diffracted light having asecond spectral peak, and the second spectral peak of the seconddiffracted light overlaps the spectral null of the first diffractedlight.
 2. The display system of claim 1, wherein the second diffractedlight has an additional spectral null and the first spectral peak of thefirst diffracted light overlaps the additional spectral null of thesecond diffracted light.
 3. The display system of claim 2, wherein thefirst hologram at least partially overlaps the second hologram in thegrating medium.
 4. The display system of claim 3, wherein the first andsecond holograms are pair-wise coherent with each other.
 5. The displaysystem of claim 1, wherein the first and second holograms are pair-wisecoherent with each other.
 6. The display system of claim 1, wherein thefirst hologram at least partially overlaps the second hologram in thegrating medium.
 7. The display system of claim 1, wherein the firsthologram comprises a first volume hologram and the second hologramcomprises a second volume hologram.
 8. The display system of claim 1,wherein the first diffracted light has side-lobe spectral peaks and thesecond diffracted light has side-lobe spectral nulls that overlap theside-lobe spectral peaks of the first diffracted light.
 9. Ahead-mounted display device comprising: a light source that produces aprobe beam; a waveguide; a grating medium on the waveguide andconfigured to receive the probe beam; a first diffractive grating in thegrating medium, wherein the first diffractive grating has a firstgrating frequency and is configured to diffract the probe beam towardsan eye box as first diffracted light having a first spectral peak and afirst spectral null; and a second diffractive grating in the gratingmedium, wherein the second diffractive grating has a second gratingfrequency that is different from the first grating frequency and isconfigured to diffract the probe beam towards the eye box as seconddiffracted light having a second spectral peak and a second spectralnull, wherein the first spectral peak overlaps the second spectral null.10. The head-mounted display device of claim 9, wherein the secondspectral peak overlaps the first spectral null.
 11. The head-mounteddisplay device of claim 9, wherein the probe beam comprises a coherentprobe beam.
 12. The head-mounted display device of claim 11, wherein thesecond diffractive grating is pair-wise coherent with the firstdiffractive grating.
 13. The head-mounted display device of claim 9,wherein the first diffractive grating comprises a first volume hologramand the second diffractive grating comprises a second volume hologram.14. The head-mounted display device of claim 9, wherein the firstdiffractive grating at least partially overlaps the second diffractivegrating in the grating medium.
 15. The head-mounted display device ofclaim 9 further comprising an output coupler on the waveguide, whereinthe output coupler comprises the first and second diffractive gratings.16. The head-mounted display device of claim 9 further comprising aninput coupler on the waveguide, wherein the input coupler comprises thefirst and second diffractive gratings.
 17. The head-mounted displaydevice of claim 9 further comprising a cross coupler on the waveguide,wherein the cross coupler comprises the first and second diffractivegratings.
 18. A head-mounted display device comprising: a waveguide; agrating medium on the waveguide; and first and second volume hologramsthat are at least partially overlapping on the grating medium, whereinthe first volume hologram is configured to diffract light incident uponthe first and second volume holograms as first diffracted light having aspectral peak and wherein the second volume hologram is configured todiffract the light as second diffracted light having a spectral nulloverlapping the spectral peak.
 19. The head-mounted display device ofclaim 18, wherein the first diffracted light has an additional spectralnull and the second diffracted light has an additional spectral peakthat overlaps the additional spectral null.
 20. The head-mounted displaydevice of claim 18, wherein the first volume hologram is pair-wisecoherent with the second volume hologram.